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# Third Isomorphism Theorem for Groups

ABSALG-2DJBTE

Let $K$, $N$ be normal subgroups of a group $G$ and let $N$ be a subgroup of $K$. Then:

$K/N$ is a normal subgroup of $G/N$

...and

$(G/N)/(K/N)\simeq G/K$

Let $G$ be a finite group with neutral element $e$ and let:

$G^n$ be the direct product of $G$ with itself $n$ times, $n\ge 1$

As a set $G^n$ is the Cartesian product of the set $G$ with itself $n$ times, and:

the group law on $G^n$ is component-wise composition in $G$

For a non-empty subset $S$ of ${\mathcal N}=\{1,\ldots,n\}$, let:

$G^S$ be the subgroup of $G^n$ consisting of all elements $(g_i)_{i=1}^n\in G^n$ with $g_t=e$, when $t\not\in S$

Let $S$, $T$ be non-empty subsets of ${\mathcal N}$.

Which of the following are true?

Select ALL that apply.

A

If $S\subseteq T$, then $(G^SG^T/G^S)/(G^T/G^S)\simeq G^S$

B

For all $S$, $T$, we have $(G^SG^T/G^{S\cap T})/(G^T/G^{S\cap T})\simeq G^{T\setminus (S\cap T)}$

C

For all $S$, $T$, we have $G^SG^T/G^T\simeq G^S/G^{S\cap T}$

D

If $S\subseteq T$, then $(G^SG^T/G^S)/(G^T/G^S)\simeq G^{(S\cup T)\setminus T}$

E

None of the above is true