The direct decompositions of a group with finitely generated
Author:
Francis Oger
Journal:
Trans. Amer. Math. Soc. 347 (1995), 19972010
MSC:
Primary 20E34; Secondary 20E07, 20F18
MathSciNet review:
1282895
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider the class which consists of the groups with finitely generated which satisfy the maximal condition on direct factors. It is well known that any group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for groups, a slightly different notion of decomposition, that we call decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of decompositions, several generalizations of results of R. Hirshon. For instance, we have for any groups , which satisfy for a group .
 [B1]
Gilbert
Baumslag, Residually finite groups with the same finite
images, Compositio Math. 29 (1974), 249–252. MR 0357615
(50 #10083)
 [B2]
Gilbert
Baumslag, Direct decompositions of finitely generated torsionfree
nilpotent groups, Math. Z. 145 (1975), no. 1,
1–10. MR
0399262 (53 #3113)
 [H1]
R.
Hirshon, Cancellation of groups with maximal
condition, Proc. Amer. Math. Soc. 24 (1970), 401–403. MR 0251130
(40 #4361), http://dx.doi.org/10.1090/S0002993919700251130X
 [H2]
Ronald
Hirshon, Some new groups admitting essentially unique directly
indecomposable decompositions, Math. Ann. 194 (1971),
123–125. MR 0286893
(44 #4100)
 [H3]
R.
Hirshon, Some cancellation theorems with applications to nilpotent
groups, J. Austral. Math. Soc. Ser. A 23 (1977),
no. 2, 147–165. MR 0447410
(56 #5722)
 [H4]
R.
Hirshon, Cancellation and Hopficity in direct products, J.
Algebra 50 (1978), no. 1, 26–33. MR 0463303
(57 #3256)
 [H5]
Ronald
Hirshon, The equivalence of
×^{𝑡}𝐶≈×^{𝑡}𝐷 and
𝐽×𝐶≈𝐽×𝐷, Trans. Amer. Math. Soc. 249 (1979), no. 2, 331–340. MR 525676
(80e:20051), http://dx.doi.org/10.1090/S00029947197905256763
 [H6]
Ronald
Hirshon, Direct decompositions of groups with finitely generated
commutator quotient group, J. Austral. Math. Soc. Ser. A
28 (1979), no. 3, 315–320. MR 557281
(81i:20036)
 [H7]
Ron
Hirshon, Some properties of direct products, J. Algebra
115 (1988), no. 2, 352–365. MR 943261
(89i:20056), http://dx.doi.org/10.1016/00218693(88)902633
 [H8]
R.
Hirshon, On uniqueness of direct decompositions of groups into
directly indecomposable factors, J. Pure Appl. Algebra
63 (1990), no. 2, 155–160. MR 1043746
(91c:20042), http://dx.doi.org/10.1016/00224049(90)90022A
 [J]
J.
M. Tyrer Jones, On isomorphisms of direct powers, Word
problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud.
Logic Foundations Math., vol. 95, NorthHolland, AmsterdamNew York,
1980, pp. 215–245. MR 579945
(82f:20068)
 [O1]
Francis
Oger, Cancellation of abelian groups of finite rank modulo
elementary equivalence, Math. Scand. 67 (1990),
no. 1, 5–14. MR 1081284
(91j:20129)
 [O2]
Francis
Oger, Cancellation and elementary equivalence of finitely generated
finitebynilpotent groups, J. London Math. Soc. (2)
44 (1991), no. 1, 173–183. MR 1122978
(92k:20003), http://dx.doi.org/10.1112/jlms/s244.1.173
 [R]
R. Remak, Uber die Zerlegung der endlichen Gruppen in direkte unzerleg bare Factoren, J. Reine Angew. Math. 139 (1911), 293308.
 [W]
R.
B. Warfield Jr., Genus and cancellation for groups with finite
commutator subgroup, J. Pure Appl. Algebra 6 (1975),
no. 2, 125–132. MR 0372053
(51 #8270)
 [B1]
 G. Baumslag, Residually finite groups with the same finite images, Compositio Math. 29 (1974), 249252. MR 0357615 (50:10083)
 [B2]
 , Direct decompositions of finitely generated torsionfree nilpotent groups, Math. Z. 145 (1975), 110. MR 0399262 (53:3113)
 [H1]
 R. Hirshon, Cancellation of groups with maximal condition, Proc. Amer. Math. Soc. 24 (1970), 401403. MR 0251130 (40:4361)
 [H2]
 , Some new groups admitting essentially unique directly indecomposable decompositions, Math. Ann. 194 (1971), 123125. MR 0286893 (44:4100)
 [H3]
 , Some cancellation theorems with applications to nilpotent groups, J. Austral. Math. Soc. (Ser. A) 23 (1977), 147165. MR 0447410 (56:5722)
 [H4]
 , Cancellation and hopficity in direct products, J. Algebra 50 (1978), 2633. MR 0463303 (57:3256)
 [H5]
 , The equivalence of and , Trans. Amer. Math. Soc. 249 (1979), 331340. MR 525676 (80e:20051)
 [H6]
 , Direct decompositions of groups with finitely generated commutator quotient group, J. Austral. Math. Soc. (Ser. A) 28 (1979), 315320. MR 557281 (81i:20036)
 [H7]
 , Some properties of direct products, J. Algebra 115 (1988), 352365. MR 943261 (89i:20056)
 [H8]
 , On uniqueness of direct decompositions of groups into directly indecomposable factors, J. Pure Appl. Algebra 63 (1990), 155160. MR 1043746 (91c:20042)
 [J]
 J. M. T. Jones, On isomorphisms of direct powers, Word Problems II, Studies in Logic 95, NorthHolland, Amsterdam, 1980, pp. 215245. MR 579945 (82f:20068)
 [O1]
 F. Oger, Cancellation of abelian groups of finite rank modulo elementary equivalence, Math. Scand. 67 (1990), 514. MR 1081284 (91j:20129)
 [O2]
 , Cancellation and elementary equivalence of finitely generated finitebynilpotent groups, J. London Math. Soc. (2) 44 (1991), 173183. MR 1122978 (92k:20003)
 [R]
 R. Remak, Uber die Zerlegung der endlichen Gruppen in direkte unzerleg bare Factoren, J. Reine Angew. Math. 139 (1911), 293308.
 [W]
 R. B. Warfield, Jr., Genus and cancellation for groups with finite commutator subgroup, J. Pure Appl. Algebra 6 (1975), 125132. MR 0372053 (51:8270)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
20E34,
20E07,
20F18
Retrieve articles in all journals
with MSC:
20E34,
20E07,
20F18
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512828958
PII:
S 00029947(1995)12828958
Keywords:
Decompositions in direct products of indecomposable groups,
cancellable in direct products,
regular,
maximal condition on direct factors
Article copyright:
© Copyright 1995
American Mathematical Society
