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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Products of involution classes in infinite symmetric groups


Author: Gadi Moran
Journal: Trans. Amer. Math. Soc. 307 (1988), 745-762
MSC: Primary 20B30
MathSciNet review: 940225
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Abstract: Let $ A$ be an infinite set. Denote by $ {S_A}$ the group of all permutations of $ A$, and let $ {R_i}$, denote the class of involutions of $ A$ moving $ \vert A\vert$ elements and fixing $ i$ elements $ (0 \leqslant i \leqslant \vert A\vert)$. The products $ {R_i}{R_j}$ were determined in [M1]. In this article we treat the products $ {R_{{i_1}}} \cdots {R_{{i_n}}}$ for $ n \geqslant 3$. Let INF denote the set of permutations in $ {S_A}$ moving infinitely many elements. We show:

(1) $ {R_{{i_1}}} \cdots {R_{{i_n}}} = {S_A}$ for $ n \geqslant 4$.

(2)(a) $ {R_i}{R_j}{R_k} = \operatorname{INF} $ if $ \{ i,\,j,\,k\} $ contains two integers of different parity; (b) $ {R_i}{R_j}{R_k} = {S_A}$ if $ i + j + k > 0$ and all integers in $ \{ i,\,j,\,k\} $ have the same parity.

(3) $ R_0^3 = {S_A}\backslash E$, where $ \theta \in E$ iff $ \theta $ satisfies one of the following three conditions:

(i) $ \theta $ moves precisely three elements.

(ii) $ \theta $ moves precisely five elements.

(iii) $ \theta $ moves precisely seven elements and has order $ 12$.

These results were announced in 1973 in [MO]. (1) and part of (2)(a) were generalized recently by Droste [D1, D2].


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0940225-9
PII: S 0002-9947(1988)0940225-9
Article copyright: © Copyright 1988 American Mathematical Society