Solving a Group Homework Problem: Finding |G/Z(G)|=20

[tyrannosaurus]

Homework Statement

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 can't use one of the nice theorems like the 2p theorem or the p^2 theorem to get the isomorphism type. Does anyone have any ideas on what I should do?

Homework Equations

The Attempt at a Solution

 

[Martin Rattigan]

Sorry to be slow. What do you want to do?

 

[tyrannosaurus]

I got that|G|=40 and |Z(G)| contains an element of order 2. The case I am having trouble us when |Z(G)|=2. Then I get |G/Z(G)| =20, and I can't use one of the nice theorems like the 2p theorem or the p^2 theorem to get the isomorphism type. i am trying to find the isomorphism type in this situation, I think it is D10 but i am not sure

 

[Martin Rattigan]

So am I right in thinking you were trying to find all possible homomorphic images of G/Z(G) given |G|=40 and Z(G) contains an element of order 2, and now you just need to find all homomorphic images given |G|=40 and |Z(G)|=2? (I'm not familiar with the term isomorphism type, but since you say it may be D₁₀, I am guessing youre talking about G/Z(G) "up to isomorphism".)

 

[Martin Rattigan]

And have you done anything about Sylow's theorems yet?

 

[Martin Rattigan]

Still not sure if we're looking at possible structures for G or G/Z (I think it's one or other), but I have to go to bed now. No doubt some kind soul will take over, otherwise I'll have a look tomorrow.

 

[tyrannosaurus]

Were looking at possible structures for G/Z. The proble is that 20 factors to 2*2*5 but we don't know if are order 20 group G/Z is D10, or a Z10+Z2 ect

 

[Martin Rattigan]

Exactly.

So the first thing, I think, would be to determine all possible groups of order 20. After that you will need to check for each of them whether they are G/Z for some possible G of order 40 with |Z(G)|=2. (Obviously for a candidate G₂₀ with 20 elements, the group Z₂xG₂₀ will have G₂₀ as a homomorphic image, but it could also have a centre larger than 2.)

Can you show first that G/Z has a normal subgroup of order 5 and a subgroup of order 4? (If you also explain your reasoning here I can guess better what you may have covered so far in your course, hence what we might reasonably use in the analysis.)