# Questions tagged [symmetric-groups]

A symmetric group is a group consisting of all permutations of given finite set, with composition of permutations as the binary operation. Should be used with the (group-theory) tag.

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### Finitary Alternating Groups

If $X$ is an infinite set, then the finitary alternating group on $X$ can be defined in the following equivalent ways: the group of all even permutations on $X$ under composition the kernel of the ...
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### Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
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### Homomorphisms from $C_2\times C_3$ to $S_4$

How many homomorphisms are there from $C_2\times C_3$ to $S_4$ are there? (Using kernel and image to describe). My thoughts/attempt: Determine homomorphisms by the image of the domain's generators. We ...
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### For α and β ∈ Sn, define α ∼ β if there exists a σ ∈ $S_n$ such that $σασ^{−1}$ = β. Show that ∼ is an equivalence relation on $S_n$
My attempt is below. Could I please get feedback on it. I am not so sure that it is correct. Let α,β,σ ∈$S_n$. Since $S_n$ is a group, we know that it contains an identity. Let e be the identity. So, \$...