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1. Hi guys, I've been set this question and I'm a bit stuck.

Find which pairs of the following groups are isomorphic:
(i) the dihedral group D6 of symmetries of a regular hexagon
(ii) Z6
(iii) Z12
(iv) the group of rotations of a regular hexagon
(v) the group of rotations of a regular tetrahedron.

I've worked out that Z6 and the rotations of a hexagon are isomorphic from comparing the Cayley tables (and they're both cyclic of order 6) but I'm not sure if there's anymore.
I know that the rotations of a tetrahedron has order 12 but I'm not sure if it's cyclic and I wouldn't know where to start with a Cayley table.

2. (Original post by Viggerz)
Hi guys, I've been set this question and I'm a bit stuck.

Find which pairs of the following groups are isomorphic:
(i) the dihedral group D6 of symmetries of a regular hexagon
(ii) Z6
(iii) Z12
(iv) the group of rotations of a regular hexagon
(v) the group of rotations of a regular tetrahedron.

I've worked out that Z6 and the rotations of a hexagon are isomorphic from comparing the Cayley tables (and they're both cyclic of order 6) but I'm not sure if there's anymore.
I know that the rotations of a tetrahedron has order 12 but I'm not sure if it's cyclic and I wouldn't know where to start with a Cayley table.

Given your instincts that the others aren't isomorphic, you'd be best looking for algebraic differences between them; two groups won't be isomorphic if;

they're of different orders.
they're not both commutative.
they're not both cyclic.
they have different numbers of elements of (a given) order n.

That should be enough to separate out these groups.
3. (Original post by RichE)
Given your instincts that the others aren't isomorphic, you'd be best looking for algebraic differences between them; two groups won't be isomorphic if;

they're of different orders.
they're not both commutative.
they're not both cyclic.
they have different numbers of elements of (a given) order n.

That should be enough to separate out these groups.
D6: order 12, commutative, non cyclic
Z6: order 6, commutative, cyclic
Z12: order 12, commutative, cyclic
rotations of hexagon: order 6, commutative, cyclic
rotations of tetrahedron: order 12, commutative, ?

and this is where I get stuck. Is what I've said correct so far?

4. (Original post by Viggerz)
D6: order 12, commutative, non cyclic
Z6: order 6, commutative, cyclic
Z12: order 12, commutative, cyclic
rotations of hexagon: order 6, commutative, cyclic
rotations of tetrahedron: order 12, commutative, ?

and this is where I get stuck. Is what I've said correct so far?

D6 is not commutative, and nor is the rotational group of a tetrahedron.
5. (Original post by RichE)
D6 is not commutative, and nor is the rotational group of a tetrahedron.
Ok, thank you.

In that case, I think that the only isomorphic pair is Z6 and the rotations of a regular hexagon.
6. (Original post by Viggerz)
Ok, thank you.

In that case, I think that the only isomorphic pair is Z6 and the rotations of a regular hexagon.
That's correct, but you still need some means of distinguishing between the two non-commutative groups of order 12.

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