G-complete reducibility in non-connected groups

In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G^0 is G^0 -cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.


Introduction
Let G be a connected reductive linear algebraic group over an algebraically closed field of characteristic 0 or characteristic p > 0. Following Serre [13], we say a subgroup H of G is G-completely reducible (G-cr) if whenever H is contained in a parabolic subgroup P of G, then H is contained in some Levi subgroup of P . The definition extends to non-connected reductive G as well: one replaces parabolic and Levi subgroups with so-called Richardson parabolic and Richardson Levi subgroups respectively (see [2], [12] and Section 2).
Even if one is interested mainly in connected reductive groups, one must sometimes consider non-connected groups. For instance, natural subgroups of a connected group, such as normalizers and centralizers, are often non-connected. The notion of G-complete reducibility is much better understood in the connected case, e.g., see [1], [9], and [10]. In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-cr. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G 0 is G 0 -cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.
An important special case of the general problem described above is the following. Let H be a subgroup of G. We say Hacts on G 0 by outer automorphisms if for each 1 = h ∈ H, conjugation by h gives a non-inner automorphism of G 0 . In this case, we may identify H with a subgroup of Out (G 0 ). Now suppose also that G 0 is simple; then H is cyclic except for possibly when G 0 is of type D 4 . It is convenient when studying conjugacy classes in G 0 to determine the fixed point set of a non-inner automorphism. See e.g., [11,Lem. 2.9] when H is cyclic and semisimple (that is, H is of order coprime to p); note that if H is generated by a semisimple element then H is G-cr by Theorem 2.5, as semisimple conjugacy classes are closed. On the other hand, if H is cyclic and unipotent (that is, H is a p-group) then H can be G-cr or non-G-cr.
We prove the following result, which gives a criterion for G-complete reducibility of H. It is an ingredient in our algorithm. In case H is cyclic, this is a special case of a recent result due to Guralnick and Malle, cf. Theorem 3.1.
Theorem (Corollary 4.5). Suppose G 0 is simple and H acts on G 0 by outer automorphisms. Then H is G-completely reducible if and only if C G 0 (H) is reductive.
Our work fits into a study begun in our earlier papers [2], [3]. It was shown in [2,Thm. 3.10] that if H is a G-cr subgroup of G and N is a normal subgroup of H then N is also G-cr. In [3] we considered a complementary question: if H is a subgroup of G, N is a normal subgroup of H and N is G-cr then under what hypotheses is H also G-cr? We gave an example (due to Liebeck) with H of the form M × N, where M and N are both G-cr but H is not [3,Ex. 5.3]. We also showed this kind of pathological behaviour does not happen when G is connected and p is good for G [3, Thm. 1.3]. Here we study the above question in the case when N is the normal subgroup H ∩ G 0 of H (see the algorithm in Theorem 5.3).

Preliminaries
2.1. Notation. Throughout, we work over an algebraically closed field k of characteristic p ≥ 0; we let k * denote the multiplicative group of k. Let H be a linear algebraic group. By a subgroup of H we mean a closed subgroup. We let Z(H) denote the centre of H and H 0 the connected component of H that contains 1. For h ∈ H, we let Int h denote the automorphism of H given by conjugation with h. Frequently, we abbreviate Int h (g) by h · g. If S is a subset of H and K is a subgroup of H, then C K (S) denotes the centralizer of S in K and N K (S) the normalizer of S in K. Likewise, if S is a group of algebraic automorphisms of H, then we denote the fixed point subgroup of S in H by C H (S). If H acts on a set X, then we also write C H (x) for the stabilizer of a point x ∈ X in H.
For the set of cocharacters (one-parameter subgroups) of H we write Y (H); the elements of Y (H) are the homomorphisms from k * to H.
The unipotent radical of H is denoted R u (H); it is the maximal connected normal unipotent subgroup of H. The algebraic group H is called reductive if R u (H) = {1}; note that we do not insist that a reductive group is connected. In particular, H is reductive if it is simple as an algebraic group. Here, H is said to be simple if H is connected and all proper normal subgroups of H are finite. The algebraic group H is called linearly reductive if all rational representations of H are semisimple.
Throughout the paper G denotes a reductive algebraic group, possibly non-connected.
Definition 2.1. Let H ⊆ G be a subgroup. We say that H acts on G 0 by outer automorphisms if for every 1 = h ∈ H, the automorphism Int h | G 0 of G 0 is non-inner, i.e., is not given by conjugation with an element of G 0 . This is equivalent to the condition that H maps bijectively onto its image under the natural map G → Aut(G 0 ) → Out(G 0 ).

G-Complete
Reducibility. In [2, §6], Serre's original notion of G-complete reducibility is extended to include the case when G is reductive but not necessarily connected (so that G 0 is a connected reductive group). The crucial ingredient of this extension is the use of so-called Richardson-parabolic subgroups (R-parabolic subgroups) of G. We briefly recall the main definitions here; for more details on this formalism, see [2, §6]. For a cocharacter λ ∈ Y (G), the R-parabolic subgroup corresponding to λ is defined by Here, for a morphism of algebraic varieties φ : k * → X, we say that lim a→0 φ(a) exists provided that φ extends to a morphism φ : k → X; in this case we set lim a→0 φ(a) = φ(0).
Then P λ admits a Levi decomposition P λ = R u (P λ ) ⋊ L λ , where We call L λ an R-Levi subgroup of P λ . For an R-parabolic subgroup P of G, the different R-Levi subgroups of P correspond in this way to different choices of λ ∈ Y (G) such that P = P λ ; moreover, the R-Levi subgroups of P are all conjugate under the action of R u (P ). An R-parabolic subgroup P is a parabolic subgroup in the sense that G/P is a complete variety; the converse is true when G is connected, but not in general ( [12,Rem. 5.3]).
and H is reductive, we can therefore associate to λ an R-parabolic subgroup of H as well as an R-parabolic subgroup of G. To avoid confusion, we reserve the notation P λ for Rparabolic subgroups of G, and distinguish the R-parabolic subgroups of H by writing P λ (H) for λ ∈ Y (H). The notation L λ (H) has the obvious meaning. Note that P λ (H) = P λ ∩ H and L λ (H) = L λ ∩ H for λ ∈ Y (H). In particular, P 0 λ = P λ (G 0 ) and L 0 λ = L λ (G 0 ). If λ ∈ Y (H) then the R-Levi subgroups of P λ (H) are the R u (P λ (H))-conjugates of L λ (H); in particular, any R-Levi subgroup of P λ (H) is of the form L ∩ H for some R-Levi subgroup L of P λ .
For later use, we record the following way to construct R-Levi subgroups. Proof. We may choose λ ∈ Y (G) such that P = P λ , P 0 = P λ (G 0 ) and M = L λ (G 0 ) = L 0 λ . We have the Levi decomposition P = R u (P λ ) ⋊ L λ = R u (P 0 λ ) ⋊ L λ . Since L λ ⊆ N P (L 0 λ ) and R u (P 0 λ ) ∩ N P (L 0 λ ) = 1 (as R u (P 0 λ ) acts simply transitively on the set of Levi subgroups of P 0 λ ), we conclude that N P (M) = N P (L 0 λ ) = L λ . Definition 2.4. Suppose H is a subgroup of G. We say H is G-completely reducible (G-cr for short) if whenever H is contained in an R-parabolic subgroup P of G, then there exists an R-Levi subgroup L of P with H ⊆ L.
Since all parabolic subgroups (respectively all Levi subgroups of parabolic subgroups) of a connected reductive group are R-parabolic subgroups (respectively R-Levi subgroups of R-parabolic subgroups), Definition 2.4 coincides with Serre's original definition for connected groups [14].
Let H be a subgroup of G and let G ֒→ GL m be an embedding of algebraic groups. Let h ∈ H n be a tuple of generators of the associative subalgebra of Mat m spanned by H (such a tuple exists for n sufficiently large). Then h is called a generic tuple of H, see [5,Def. 5.4]. We recall the following geometric criterion for G-complete reducibility [5,Thm. 5.8]; it provides a link between the theory of G-complete reducibility and the geometric invariant theory of reductive groups. The following result has been proved with methods from geometric invariant theory (see [5,Def. 5.17]): Theorem 2.6. Assume that the subgroup H of G is not G-completely reducible. Then there exists an R-parabolic subgroup P of G with the following two properties: The geometric construction of P in [5, §4] is roughly as follows: there is a class of so-called optimal destabilizing cocharacters Ω ⊆ Y (G) such that if λ ∈ Ω then P := P λ has properties (i) and (ii) as in Theorem 2.6. We call such an R-parabolic subgroup P of G an optimal destabilizing R-parabolic subgroup for H.

2.3.
Criteria for G-complete reducibility. In this subsection we study criteria for Gcomplete reducibility in terms of some smaller group.
A homomorphism π : G 1 → G 2 is called non-degenerate provided that ker(π) 0 is a torus. The next result is contained in [2, Lem. 2.12 and §6]: As an immediate consequence, we obtain the following result which allows us to focus on the part of G that is effectively acting on G 0 . Note that The following lemma gives two necessary conditions for a subgroup of G to be G-completely reducible, both of which can be checked in the connected group G 0 . Lemma 2.10. Let H be a G-completely reducible subgroup of G. Then the following hold: Proof. (i). This is the content of [2, Lem. 6.10 (ii)]. (ii). Since H is G-cr, so is its centralizer Under the assumption that assertion (i) or (ii) of Lemma 2.10 holds, the next two lemmas allow us to replace the ambient group G with a potentially smaller subgroup M.
Lemma 2.11. Let H be a subgroup of G and suppose that H ∩G 0 is G 0 -completely reducible.
, it is reductive, by [13,Property 4]. The same is true for H ∩ G 0 by assumption. Hence (H ∩ G 0 )C G 0 (H ∩ G 0 ) is the product of two reductive groups and thus is reductive. As this group contains M 0 as a normal subgroup, the group M is reductive as well.
be an optimal destabilizing cocharacter for H in G. Then P λ (G 0 ) contains the subgroup H ∩ G 0 , which is G 0 -cr by assumption. Hence after replacing λ with an R u (P λ )-conjugate, we may assume that λ centralizes H ∩ G 0 . This implies that λ ∈ Y (M), and H ⊆ P λ (M) ⊆ P λ . Since H is not contained in an R-Levi subgroup of P λ (cf. Theorem 2.6), it is not contained in an R-Levi subgroup of P λ (M). We conclude that H is not M-cr.
Proof. We proceed as in the proof of Lemma 2.11: Again, M is well-defined since H normalizes by assumption, as before we may conclude that its centralizer is reductive, so that M is reductive.
Suppose that H is not G-cr. Let λ ∈ Y (G) = Y (G 0 ) be an optimal destabilizing cocharacter for H in G. By Theorem 2.6(ii), P λ contains C G 0 (H), which is G 0 -cr. Thus we may again assume that λ centralizes C G 0 (H), so that λ ∈ Y (M). As before, we conclude that H is not M-cr.
. As before, this shows that H is M-cr.
Remark 2.14. Let H be a subgroup of G and let π : G → G ′ be an isogeny. Then π( We may write the connected reductive group G 0 in the form where S is the radical of G 0 and G 1 , . . . , G n are the simple components of the derived group of G 0 . Any subgroup H of G acts via conjugation on the derived subgroup of G 0 and hence permutes the simple components. We obtain an induced action of H on the set of indices {1, . . . , n}. For 1 ≤ i ≤ n, we use the shorthand for the product of all factors in G 0 above with the exception of G i . Our next lemma allows us to replace G with a collection of reductive groups whose identity components are simple.
Proof. First note that, by construction, H i and G 0 both normalize the group G i . Hence the map π i is well-defined. Since H i G 0 is reductive, so is its image under π i .
To prove the forward implication, suppose the assertion fails for some i ∈ I. Up to reordering the indices, we may assume that π 1 (H 1 ) is not π 1 (H 1 G 0 )-cr and that H acts transitively on the set {1, . . . , r} for some r ≥ 1. Let Q be an optimal destabilizing Rparabolic subgroup of π 1 (H 1 G 0 ) for π 1 (H 1 ). To obtain a contradiction, we show that π 1 (H 1 ) is contained in an R-Levi subgroup of Q. Consider the group Q 0 . This is a parabolic subgroup of Since P 1 contains the centre of G 1 and π 1 (P 1 ) = Q 0 is normalized by π 1 (H 1 ) ⊆ Q, it follows that P 1 is normalized by H 1 . Indeed, let h ∈ H 1 . Then h · P 1 ⊆ G 1 . On the other hand, π 1 (h · P 1 ) = π 1 (h) · π 1 (P 1 ) = π 1 (P 1 ). Since ker(π 1 ) = G 1 , this implies that h · P 1 ⊆ P 1 G 1 . We conclude that h · P 1 ⊆ P 1 (G 1 ∩ G 1 ) = P 1 , where we have used that the last intersection is central in G 1 .
For 2 ≤ j ≤ r, let h j ∈ H be an element satisfying h j · G 1 = G j . Let P j = h j · P 1 , which is a parabolic subgroup of G j . Since we have just verified that H 1 normalizes P 1 , the definition of P j does not depend on the choice of h j that transports G 1 to G j .
We now consider the parabolic subgroup P = SP 1 · · · P n of G 0 , where we take P j = G j for j > r. By construction, P is normalized by H. Indeed, any h ∈ H fixes S under conjugation, and permutes the groups G 1 , . . . , G n . If h maps G i to G j with i, j ∈ {1, . . . , r}, then (hh i ) · G 1 = G j , and hence h · P i = (hh i ) · P 1 = P j . So h also permutes the groups P 1 , . . . , P r , and thus normalizes P .
The group N G (P ) is thus an R-parabolic subgroup of G containing H with N G (P ) 0 = P (see [2, Prop. 6.1]). Since H is G-cr, it is contained in an R-Levi subgroup L of N G (P ), hence it normalizes the Levi subgroup L 0 of P . We may write L 0 = SL 1 · · · L n for certain Levi subgroups L j of P j . Then H 1 normalizes L 1 , since L 1 = L 0 ∩ G 1 . This forces π 1 (H 1 ) to normalize a Levi subgroup of Q 0 = π 1 (P 1 ). By Lemma 2.3, π 1 (H 1 ) is contained in an R-Levi subgroup of Q, yielding a contradiction.
To prove the reverse implication, we again assume after reordering the indices that 1 ∈ I and that H permutes the set {1, . . . , r} transitively for some r ≥ 1. Assume that H is not G-cr, and that Q ⊆ G is an optimal destabilizing R-parabolic subgroup of G containing H. Again we want to deduce that H is contained in an R-Levi subgroup of Q, contradicting our assumption.
Since L 1 contains the centre of G 1 , as in the proof of the forward implication (where we have proved that H 1 normalizes P 1 ), we may conclude that H 1 normalizes L 1 . Choosing again elements h j ∈ H with h j · G 1 = G j for 2 ≤ j ≤ r, we obtain well-defined Levi subgroups L j := h j · L 1 of h j · P 1 = P j , where the latter equality follows from P j = Q 0 ∩ G j . Proceeding similarly for the other H-orbits on {1, . . . , n} (each of which contains an element of I by assumption), we construct an H-stable Levi subgroup L = SL 1 · · · L n of Q 0 . As before, by Lemma 2.3, H is contained in an R-Levi subgroup of Q, which gives the desired contradiction. This finishes the proof.  Combining some of our previous reductions, we obtain the following weaker version of Theorem 3.1. This is of independent interest, as our arguments allow us to avoid the caseby-case considerations that are needed for the proof of Theorem 3.1. Proof. The forward implication is clear, by Lemma 2.10(ii). Conversely, assume that C G 0 (H) is reductive. Since linearly reductive subgroups are G-cr (see [2,Lem. 2.6]), we may assume that k has positive characteristic p that coincides with the order of H.

Cyclic subgroups
We first show that C G 0 (H) is G 0 -cr. Suppose this fails, and let P ⊆ G 0 be an optimal destabilizing parabolic subgroup for C G 0 (H) in G 0 . Then H normalizes P , by Theorem 2.6(ii). Let U be the unipotent radical of P . Then Z(U) has positive dimension and is normalized by H and P . Up to passing to a characteristic subgroup (the subgroup of elements of order dividing p), we may assume that Z(U) has exponent p. Thus Z(U) has the structure of an F p -vector space of infinite dimension with an F p -linear H-action. As H is generated by an element h ∈ H of order p, there must be infinitely many fixed points of H on Z(U). Indeed, on any H-stable finite dimensional subspace W of Z(U) the automorphism induced by h may be brought into Jordan normal form with block sizes bounded by p (the Jordan normal form exists as h has only eigenvalue 1 ∈ F p ). As each block contributes at least p − 1 fixed points, H has at least (p − 1)[dim W/p] fixed points on W , and we can make dim Fp W arbitrarily large. Taking the identity component of the H-fixed points on Z(U) hence yields a non-trivial, connected, normal, unipotent subgroup of C G 0 (H), contradicting the reductivity assumption. We thus conclude that C G 0 (H) is G 0 -cr.
By Lemma 2.13, it therefore suffices to show that H is M-cr, where M = HC G 0 (C G 0 (H)). Let M 1 , . . . , M r be the simple components of M 0 . By definition of M, C M i (H) ⊆ M i ∩Z(M 0 ) is finite for each i. This forces H i = N H (M i ) = 1 for each i. Indeed, since H has prime order, H i = 1 would yield that H normalizes M i , but due to a result of Steinberg (cf. [16,Thm. 10.13]), no non-trivial cyclic group can act on a simple group via algebraic automorphisms with only finitely many fixed points. By Lemma 2.16, we conclude that H is M-cr, as required.

Outer automorphisms for D 4
In this section, let D 4 denote an adjoint simple group of type D 4 . Amongst the simple groups D 4 has the largest outer automorphism group, in that Out(D 4 ) ∼ = S 3 , the symmetric group on 3 letters. We may identify Out(D 4 ) with the set of graph automorphisms in Aut(D 4 ) induced by the symmetries of the Dynkin diagram. However, there are other subgroups isomorphic to S 3 in Aut(D 4 ) that act via outer automorphisms. As this is the only situation where outer automorphisms of a simple group arise that is not covered by Theorem 3.3, we treat this case separately in this section.
Let T be a maximal torus of D 4 with associated root system Φ. Let ∆ = {α, β, γ, δ} be a set of simple roots for Φ, where δ is the unique simple root that is non-orthogonal to every other simple root. Let λ = ω ∨ δ ∈ Y (T ) be the fundamental dominant coweight determined by α, λ = β, λ = γ, λ = 0, δ, λ = 1. For ǫ ∈ Φ we denote by u ǫ : G a → U ǫ a fixed root homomorphism onto the corresponding root subgroup of G.
Then C D 4 (σ) is a simple group of type G 2 . In fact,T = C T (σ) is a maximal torus of C D 4 (σ), andα = α|T = β|T = γ|T andβ = δ|T form a pair of simple roots with respect toT , with corresponding root groups given by uα(c) = u α (c)u β (c)u γ (c), uβ(c) = u δ (c). Since λ evaluates inT , we may regard it as an element of Y (T ); we denote this element byλ. We have α,λ = 0, β ,λ = 1. We begin with a detailed description of triality in the particular case where the ground field has characteristic three, using the results of [6] and [7]. Proposition 4.1. Assume that p = 3. In Aut(D 4 ) there are exactly two conjugacy classes of cyclic groups of order three generated by outer automorphisms. Let σ 1 , σ 2 be representatives of the respective classes, and let M i = C D 4 (σ i ) (i = 1, 2). Then we may choose the labelling such that the following holds: (i) M 1 is a simple group of type G 2 ; moreover Aut(D 4 ) · σ 1 , the orbit of σ 1 under conjugation, is closed in Aut(D 4 ). (ii) M 2 is an 8-dimensional group with 5-dimensional unipotent radical and corresponding reductive quotient isomorphic to SL 2 ; the orbit Aut(D 4 )·σ 2 is not closed and contains σ 1 in its closure.
(v) With the choices in (iii) and (iv), we have where Pλ denotes P λ (M 1 ).
Proof. By [6, Cor. 6.5, Thm. 9.1], there are precisely two conjugacy classes of cyclic groups of order three generated by outer automorphisms, which are denoted by type I and type II, respectively. They are distinguished by the structure of the corresponding fixed point groups, where type I yields a group of type G 2 , whereas type II in characteristic 3 gives a group with the structure described in (ii) (see [6, §9] together with [7,Thm. 7]). This implies the first statements of (i) and (ii), as well as (iii). Working in the algebraic group Aut(D 4 ), using σ(λ) = λ and α + β + γ + 2δ, λ = 2 > 0 we compute that lim To prove (v), we first note that R u (Pλ) consists of the root groups for the rootsβ,α +β, 2α +β, 3α +β and 3α + 2β. In particular, the semi-direct product Uα, U −α ⋉ R u (Pλ) has dimension 8 and is contained in C M 1 (u) = C M 1 (u 3α+2β (1)). Asλ centralizes ±α, the semi-direct product is also contained in Pλ. Since clearly C M 1 (u) ⊆ M 2 , the assertion (v) follows by comparing dimensions. This finishes the proof.
We can now characterize G-complete reducibility in the case where G 0 = D 4 and H maps isomorphically onto the full group of outer automorphisms of D 4 . The following result is the analogue of Theorem 3.3. Proof. The forward implication is clear by Lemma 2.10. Conversely, assume that H is not G-cr. Let h ∈ H be an element of order 3, so that K = h is a normal subgroup of index 2 in H. By the assumption on H, the map π : G → Aut(D 4 ), g → Int(g)| G 0 is surjective. Since ker(π) = C G (G 0 ), π is an isogeny. Hence π(H) is not Aut(D 4 )-cr, by Lemma 2.8. It now follows from Remark 2.14 that we can take G to be Aut(D 4 ).
First assume that p = 3. Let M 1 = C D 4 (σ) and M 2 = C D 4 (σu) with notation as in Proposition 4.1. Then K is a normal subgroup of order 3 and index 2 in H. Since p = 3 is coprime to 2, we have by Theorem 2.7(ii) that K is not Aut(D 4 )-cr. This implies (by Theorem 2.5) that the orbit Aut(D 4 ) · h is not closed, whence by Proposition 4.1 there exists g ∈ G with ghg −1 = σu. Replacing H with gHg −1 , we may assume that h = σu. Let s ∈ H be an element of order 2 such that h and s generate H. Let τ ∈ Aut(D 4 ) be the graph automorphism of order 2 determined by s, i.e., the graph automorphism that induces the same element as s in Out(D 4 ). Let t =β ∨ (−1) ∈T ⊆ M 1 . Since τ and σ fix M 1 , both elements commute with t and u. Moreover, by construction tut = u −1 . This implies that τ t has order 2 and (τ t)(σu)(τ t) = τ στ u −1 = σ −1 u −1 = (σu) −1 . As τ t induces the same element as s in Out(D 4 ), we can find x ∈ D 4 with s = τ tx. We conclude that both pairs of elements h = σu, s = τ tx as well as σu, τ t generate a group isomorphic to S 3 . In particular, x(σu)x −1 = (τ t)(τ tx)(σu)(τ tx) −1 (τ t) = (τ t)(σu) −1 (τ t) = σu. Thus x ∈ M 2 ⊆ M 1 (cf. Proposition 4.1(v)), so that s = τ tx normalizes M 1 . As M 1 is simple of type G 2 , it has no outer automorphisms. Therefore we may find s ′ ∈ M 1 with Int(s)| M 1 = Int(s ′ )| M 1 . Since M 1 is adjoint, s ′ is of order 2. Now Since s = τ tx normalizes M 2 , s normalizes N M 1 (M 2 ) = Pλ (see Proposition 4.1(v)). Hence s ′ ∈ Pλ and C M 2 (s ′ ) ⊆ Pλ. Up to conjugation in Pλ we may thus assume s ′ ∈T . AsT is generated by the images ofα ∨ andβ ∨ , this reduces the possibilities to s ′ ∈ {α ∨ (−1),β ∨ (−1),α ∨ (−1)β ∨ (−1)}. But then s ′ centralizes U 3α+2β , or Uβ, or Uα +β respectively. We deduce that is not reductive, as required. Now let p = 3. Then the subgroup K of H of order 3 is linearly reductive, in particular it is G-cr and C G 0 (K) is reductive. Moreover, the group C G 0 (K) is connected being the fixed point group under a triality automorphism (cf. [6, §9]). Let M = HC G 0 (K). By [3, Thm. 3.1(b)(ii)] applied to K ⊆ H ⊆ M, we deduce that H is not M-cr. Since K is normal in M, by Theorem 2.7(i), H/K is not M/K-cr. But H/K is cyclic of order 2, so we may apply Theorem 3.3 to conclude that C (M/K) 0 (H/K) is not reductive. By construction, This finishes the proof.
Having settled the case of D 4 , we can combine Theorems 3.3 and 4.2 to characterize Gcomplete reducibility in case G 0 is simple and the subgroup H acts by outer automorphisms. Proof. We may assume that G 0 is adjoint (cf. the first paragraph of the proof of Theorem 4.2). Since H acts via outer automorphisms, we may identify it as an abstract group with a subgroup of Out(G 0 ), the finite group of outer automorphisms of G 0 . As G 0 is simple, Out(G 0 ) is either simple of prime order or G 0 is of type D 4 and Out(G 0 ) ∼ = S 3 . The result now follows from Theorems 3.3 and 4.2.

The Algorithm
We return to the general situation where H ⊆ G is a subgroup of a possibly non-connected reductive group. In this section, we are going to establish an algorithm that reduces the question of whether H is G-cr to the question of whether certain subgroups of certain connected reductive groups are G-completely reducible.

From
Step 3 on, we may assume in addition that p ∈ {2, 3} and that H is not contained in G 0 . The conclusion of Step 3 is correct by Lemma 2.10(i).
Step 4 is an application of Corollary 4.5.
Since we have passed Step 3, we may assume that H ′ ∩ G ′0 is G ′0 -cr. Under the condition of Step 5, H ′ /(H ′ ∩ G ′0 ) is cyclic of prime order. The conclusion of Step 5 thus follows from Remark 2.12 and Theorem 3.3.
Finally, Step 6 is again covered by Proposition 5.1. Moreover, this step is only applicable for G ′0 simple of type D 4 . As we may assume H ′ ∩ G ′0 = 1 and Z(G ′0 ) = 1, the group It remains to show that the algorithm terminates.
Step 1 may restart finitely many instances of the algorithm. In each instance the algorithm terminates in Step 2 -Step 5 if Step 6 is not reached. If Step 6 is applicable, it replaces G ′ -which is simple of type D 4 -with a group of smaller dimension. This implies that after Step 1 is applied again, Step 6 cannot be reached a second time, and the algorithm terminates.
Remark 5.4. (i). It follows from the proof of Theorem 5.3 that Step 1, the only step that replaces a pair with several new pairs, need only be done at most twice along a path through the algorithm. Also, Step 6 only occurs at most once.
(ii). There are some situations where shortcuts may be applied to reduce to a connected group. First of all, if H 0 is not reductive, then H cannot be G-cr. On the other hand, if H is cyclic, then we may apply Theorem 3.1 to deduce that H is G-cr if and only if C G 0 (H) is G 0 -cr. Finally, if H/(H ∩ G 0 ) is linearly reductive, we can apply Theorem 2.7(ii) to deduce that H is G-cr if and only if H ∩ G 0 is G 0 -cr. However, the proposed algorithm gives a systematic approach that deals with all possible cases.
(iii). If p = 0, then a subgroup H is G-cr if and only if it is reductive ( [14,Prop. 4.2]). Of course, H is reductive if and only if H 0 is reductive, which in turn is equivalent to H 0 being G 0 -completely reducible.

Examples
We conclude with some examples of the algorithm outlined in Theorem 5.3. Example 6.1. Let p = 3, G = Aut(D 4 ). Let σ be the triality graph automorphism as in Section 4. Let H = σ K, where K = C D 4 (σ) is the fixed point subgroup of type G 2 . We follow through the algorithm to deduce that H is G-cr: Step 1 is not applicable, as G 0 = D 4 is simple and C G (G 0 ) = 1. In Step 2 we obtain n = 3 = p as the order of σ ∼ = H/(H ∩ G 0 ). Now H ∩ G 0 = K is G-cr (see Corollary 3.2), hence Steps 3 and 4 are not applicable.
Step 5 applies and leads us to consider the group M = HC D 4 (K)/K ∼ = σ C D 4 (K). As K is adjoint, we obtain C M 0 (σ) = 1 and thus this group is clearly M 0 -cr. The algorithm stops with the conclusion that H is G-cr.
Here we have two commuting G-cr subgroups σ and K of G and their product is also G-cr. This is not always the case: see [3, Ex. 5.1].
Example 6.2. Let Γ be a finite group acting transitively on a finite set I. Let i 0 ∈ I. Let ρ : Γ → M be a homomorphism to a simple group M such that ρ(C Γ (i 0 )) is not M-cr. We set G = Γ ⋉ i∈I M,