# Math Help - isomoprhism

1. ## isomoprhism

Hi,

whcih of the following groups would be isomorphic?

D32
A5
D22
A22
A32
S32
S22
Z60
S60
A60.

thanks

2. Originally Posted by nidak
D32
64 elements

A5
60 elements.

D22
44 elements.

A22
22!/2 elements.

A32
32!/2 elements.

S32
32! elements

S22
22! elements.

Z60
60 elements.

S60
60! elements.

A60.
60!/2 elements.

The only ones that have a chance of being isomorphic are A5 and Z60. But A5 is non-abelian and Z60 is. So there is no isomorphism here.

3. ## more..

C x R
Z4 x Z3 x Z5
D11 x Z2
Z4 x Z8
S22
Z60
A5
A22
A32
S32
A60
S60

4. Originally Posted by nidak
C x R
$2^{\aleph_0}$
Z4 x Z3 x Z5
60
D11 x Z2
44
Z4 x Z8
32
S22
22!
Z60
60
A5
60
A22
22!/2
A32
32!/2
S32
32!
A60
60!/2
S60
60!

The only possible ones are:
Z4xZ3xZ5 and Z60 and A5
We note the first two are abelian and third one is not.
Thus, it remains to show that the first two are or not isomorphic.

Indeed they are because 4,3,5 in Z4xZ3xZ5 are relatively prime.