# Math Help - isomorphism type question

1. ## isomorphism type question

I got that|G|=40 and |Z(G)| contains an element of order 2. From Lagrange i know that the order of Z(G) must divide |G| and be a multiple of 2. I am able to do all the cases by the G/Z theorem accept for 1 case. This is the case where |Z(G)|=2. Then I get |G/Z(G)| =20, and I cant use one of the nice theorms like the 2p theorem or the p^2 theorem to get the isomorphism type. Does anyone have any ideas on what I should do?

2. Originally Posted by nhk
I got that|G|=40 and |Z(G)| contains an element of order 2. From Lagrange i know that the order of Z(G) must divide |G| and be a multiple of 2. I am able to do all the cases by the G/Z theorem accept for 1 case. This is the case where |Z(G)|=2. Then I get |G/Z(G)| =20, and I cant use one of the nice theorms like the 2p theorem or the p^2 theorem to get the isomorphism type. Does anyone have any ideas on what I should do?
Off the top of my head, it has even order and a centre of order 2 - surely it is just $D_{40}$? Certainly, this is a group with the correct properties. There may be other groups with these properties though, I've not quite got enough time to look at this properly. Sorry!