Re: D_8 and Q_8

From: Ron Sperber (ronsperber_at_optonline.net)
Date: 08/31/04 

Date: Tue, 31 Aug 2004 03:25:50 GMT

Isaac wrote:
> Any easy way to see these are not isomorphic other than "my way" which is
> writing out all of the elements of each and seeing that there are more order
> 4 elements in Q_8 than in D_8? I ask because the question would be harder
> if the groups in question were of higher order, and thus I wouldn't be able
> to write down all the elements.
>
> Isaac
>
>
What do you mean by "easy way"? Determining that 2 groups are NOT
isomorphic requires finding some property that one has that other
doesn't. Clearly for finite groups, one possibility would be # of
elements. This doesn't help for Q_8 vs. D_8 since they both have 8
elements. Another might be if one is abelian and the other isn't. Again
this fails. So you have to try something else. In the case of D_8 vs.
Q_8, you could count elements of order 4, or you could note that D_8 is
isomorphic to the semidirect product of Z_4 and Z_2, while Q_8 cannot be
written that way. In general, given presentations for 2 groups G_1 and
G_2, there is no good algorithm to determine whether or not they are
isomorphic.

  -Ron