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Abstract Algebra

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Moderate

Subgroups of S_4 under Homomorphism

ABSALG-YJEJ4H

Let $S_4$ be the group of permutations of the set $T_4=\{1,2,3,4\}$.

Let $T_3$ be the set of partitions of $T_4$ of the form:

$$\{1,2\}\cup\{3,4\},\quad \{1,3\}\cup\{2,4\},\quad \{1,4\}\cup\{2,3\}$$

Let $S_3$ be the group of permutations of the $3$ elements of $T_3$.

Define the homomorphism $f:S_4\rightarrow S_3$ by letting $f(\sigma)$, $\sigma\in S_4$, be the action:

$\{i,j\}\cup\{k,\ell\}\rightarrow \{\sigma(i),\sigma(j)\}\cup\{\sigma(k),\sigma(\ell)\}, \quad i,j,k,\ell$ mutually distinct

...on $T_3$.

Which of the following are true?​ Select ALL that apply.

A

The image of $\{1,(123),(132)\}$ under $f$ is an abelian subgroup of $S_3$

B

The kernel of $f$ is a normal non-abelian subgroup of $S_4$

C

The image of $\{1,(12),(34),(12)(34)\}$ under $f$ has order $2$

D

$f((1234))$ generates a cyclic subgroup of $S_3$ order $3$

E

The cardinality of the set $f^{-1}(\langle f((1234))\rangle)$ is $4$, where:

$\langle f((1234))\rangle$ is the cyclic subgroup of $S_3$ generated by $f((1234))$.