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Abstract Algebra

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Moderate

Subgroups: From Sets, Conjugates, Unions, Intersections in $S_4$

ABSALG-MSECJW

Let $S_4$ be the symmetric group on $4$ symbols, and let $e$ be the neutral (identity) element of $S_4$.

Let $A_4$ denote the alternating on $4$ symbols, $V_4$ the Klein $4$-group, and $D_4$ the dihedral group with $|D_4|=8$).

We say two subgroups of $S_4$ are equal if and only if they are isomorphic to each other.

For a subset $T\subseteq S_4$, let $\langle T\rangle$ denote the subgroup of $S_4$ generated by $T$.

To ease notation, we often omit the set brackets in the description of $T$.

For $\sigma$ in $S_4$, let $\sigma T\sigma^{-1}=\{\sigma t \sigma^{-1}\mid t\in T\}$.

1) $\langle (12)(34)\rangle$ equals

.

2) $\langle (12)\rangle$ equals

.

3) The set $\bigcup_{\sigma\in S_4}\sigma\{e,(12)(34)\}\sigma^{-1}$ equals

, which is

in $S_4$.

4) The set $\bigcup_{\sigma\in S_4}\sigma\{e,(12)\}\sigma^{-1}$ equals

, which is

in $S_4$.

5) The set $\bigcap_{\sigma\in S_4}\sigma\langle (12), (12)(34)\rangle\sigma^{-1}$ equals

.

6) The set $\bigcap_{\sigma\in S_4}\sigma \langle (1234),(13) \rangle\sigma^{-1}$ equals

.