# Isomorphic groups

• Oct 31st 2010, 08:34 PM
jzellt
Isomorphic groups
Here is a set of four groups: Z4, Z2 x Z2, P2, V
Determine which groups are isomorphic to which others.

* P2 denotes the group of subsets of a two element set
* V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.

Thanks for the help!!
• Oct 31st 2010, 08:39 PM
Drexel28
Quote:

Originally Posted by jzellt
Here is a set of four groups: Z4, Z2 x Z2, P2, V
Determine which groups are isomorphic to which others.

* P2 denotes the group of subsets of a two element set
* V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.

Thanks for the help!!

How much group theory do you know? Since $4=2^2$ all groups of order four are abelian. Then, by the structure theorem any group of order four is isomorphic to $\mathbb{Z}_4$ or $\mathbb{Z}_2\oplus\mathbb{Z}_2$ according to whether their is an element of order $4$ or not.
• Oct 31st 2010, 08:39 PM
tonio
Quote:

Originally Posted by jzellt
Here is a set of four groups: Z4, Z2 x Z2, P2, V
Determine which groups are isomorphic to which others.

* P2 denotes the group of subsets of a two element set
* V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.

Thanks for the help!!

Well, the first group is the cyclic one of order 4, and the second one is the Klein group (non-cyclic

of order 4), so what kind of groups of order 4 are the 3rd and 4th ones?

Tonio