# how to show that the groups are not isomorphic

$Z_2\times Z_2\times Z_3$ , $Z_4 \times Z_3$ , $D_{12}$, $A_4$

Show that no 2 groups are isomorphic to each other?

$Z_2 \times Z_2 \times Z_3$ and $Z_4 \times Z_3$): For $Z_2 \times Z_2 \times Z_3$ and $Z_4 \times Z_3$ I said that they are not isomorphic because $Z_2 \times Z_2 \times Z_3$ has an LCM of 6 vs. 12 so they aren't isomorphic. Am I right?

$Z_2 \times Z_2 \times Z_3$ and $D_{12}$: I am not sure, but I think because the order of $Z_2 \times Z_2 \times Z_3$ is 6 and $D_{12}$ is 12, they are not isomorphic?

$Z_2 \times Z_2 \times Z_3$ and $A_4$: ? Not sure?

$Z_4 \times Z_3$ and $D_{12}$: ?

$Z_4 \times Z_3$ and $A_4$: ?

$D_{12}$ and $A_4$: I said they aren't isomorphic because $D_{12}$ has a rotation of order 12 and $A_4$ has order 1, 2 or 3 so they are not. ( Is this correct?)

I'd appreciate if you can tell me if I am right / wrong for the ones I solved and help me solve the remaining ones. Thanks!

• The order of $\mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/3$ is $2\cdot 2\cdot 3=12$, not $6$. Commented Nov 18, 2016 at 0:22
• For your LCM argument, why does the LCM matter? $D_{12}$ does not have an element of order $12$, I think (from context) that it has an element of order $6$, i.e., the dihedral group of order $12$. Commented Nov 18, 2016 at 0:25
• Why does it make sense to compare the lcm of $\{2,2,3\}$ with $12$ for $D_{12}$. The LCM is the maximum order of an element in a product of cyclic groups while the $D_{12}$ refers to the order of the group. Commented Nov 18, 2016 at 0:26
• DO NOT VANDALIZE your own post, miller21! Commented Nov 18, 2016 at 0:54

$D_{12}$ has an element of order $6$ but $A_4$ doesn't and so they are not isomorphic.