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Infinite symmetric groups. (English) Zbl 1281.20003

The author provides proofs of several well-known trivial facts related to the infinite symmetric group. Namely, let \(S(m)\) be the group of all permutations of the set \(\{1,2,\dots,m\}\subset\mathbb N\). The group \(S(m)\) acts on \(\mathbb N\) fixing all integers greater than \(m\). In Proposition 2 the author shows that the union \(\bigcup S(m)\) is a group. This group is sometimes called the infinite symmetric group \(S(\infty)\). Denote \(S_m(\infty)=\{s\in S(\infty):s(j)=j\text{ for all }j\leq m\}\), \(B(m)=\{s\in S(\infty):s(j)\leq j\text{ for all }j\leq m\}\). Theorem 1 states that \(B(m)=S(m)\times S_m(\infty)\). In Section 2.4 the author describes the matrix presentation for symmetric groups. Consider the equivalence relation on the group \(G\) of all bijections \(\mathbb N\to\mathbb N\) given by: \(\alpha\sim\beta\Leftrightarrow\alpha(j)=\beta(j)\) for sufficiently large \(j\). Section 3 is devoted to proving the fact that \(G/\sim\) is a group.

MSC:

20B30 Symmetric groups
20B35 Subgroups of symmetric groups
20C32 Representations of infinite symmetric groups
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