MHB Finding Isomorphisms between Groups: D6, A4, S3xZ2, G

[Enzipino]

I'm having a bit of problem trying to find isomorphisms between the following groups:
$D_6$, $A_4$, $S_3 \times \Bbb{Z}_2$, and $G$.

• G is a group generated by $a, b, c$ which follow these rules: $a^2=b^2=c^3=id$ (id = identity), $ca=bc$, $cb=abc$, $ab=ba$.

I can find isomorphisms between basic $\Bbb{Z}'s$ just fine but once I get to these types of groups I come to a complete stop. I know I have to consider the order of their respective elements but I just don't know where to go after that. Could anyone help me at least start one pair?

 

[I like Serena]

I'm having a bit of problem trying to find isomorphisms between the following groups:
$D_6$, $A_4$, $S_3 \times \Bbb{Z}_2$, and $G$.

• G is a group generated by $a, b, c$ which follow these rules: $a^2=b^2=c^3=id$ (id = identity), $ca=bc$, $cb=abc$, $ab=ba$.

I can find isomorphisms between basic $\Bbb{Z}'s$ just fine but once I get to these types of groups I come to a complete stop. I know I have to consider the order of their respective elements but I just don't know where to go after that. Could anyone help me at least start one pair?

Hi Enzipino,

To disprove the existence of an isomorphism, an easy check is to look at the orders of the generators.
If they are different, there can't be an isomorphism.

For instance $D_6$ has a generator of order 6, while the highest order of an element in $A_4$ is 3.
Therefore there is no isomorphism.

To find isomorphisms, it's usually easiest to look at the multiplication table of just the generators.
If there is a match, we have an isomorphism.
What are the generators of your groups?