Hello!

Using the external direct product $\displaystyle \oplus$, the group

$\displaystyle S_3 \oplus Z_2$ is isomorphic to one (and only one!) of the following groups (this question reminds me of a game show...)

a) $\displaystyle Z_{12}$

b) $\displaystyle Z_6 \oplus Z_2$

c) $\displaystyle A_4$

d) $\displaystyle D_6$

I am assuming these groups are the well known groups

$\displaystyle Z_n$ are the integers modulo n under addittion

$\displaystyle A_n$ is the alternating group (all even permutations of n) under function composition

$\displaystyle D_n$ is the dihedral group of order $\displaystyle 2n$

and $\displaystyle S_n$ is the set of permutations of n

The question recommends determining which groups it is not isomorphic to and determining the answer by elimination.

I started by the question noting that $\displaystyle S_3 \oplus Z_2$ is cyclic, I believe? so it must be isomorphic to a cyclic group as well...

and also it must have the same number of elements of each order of any group it is isomorphic... but even armed with those facts I am having trouble finding that any of a), b), c) or d) are isomorphic to the given group $\displaystyle S_3 \oplus Z_2$

Any help appreciated!

Thank you!