From: mckay@cs.concordia.ca (MCKAY john) Subject: Re: Question of the concept isomorphism of group theory Date: 22 Dec 1999 11:13:51 GMT Newsgroups: sci.math Keywords: Brauer pair (groups indistinguishable by character theory) In article <83q5n9\$9lk\$1@nnrp1.deja.com> jrbao@my-deja.com writes: >I have this basic and not too difficult question, but I haven't very >clear idea. I hope someone can give me an explaination. > >We know, if two group A and B are isomorphic, they have the same >character table. My question is: How about the inverse case? i.e. if two >groups have the same character table, are they isomorphic? > NO - it is not. Even the stronger property that the char tables AND the power maps (induced on classes by the maps g -> g^k, g in G) can correspond and yet the groups are not pairwise isomorphic. Such groups are called a Brauer pair. There is a Brauer pair of order 2^8. They are subgroups of a common index 2 supergroup. They have a common index 2 subgroup. They have a faithful permutation representation of degree 32. I do not know whether a minimal such pair is necessarily a p-group. I believe the smallest order pair is of order 2^8. John McKay >I have this question because of two groups of order 8: Quaternion group >Q8 and Dihedral group D8 (sometimes writing as D4). As I know they are >nonisomorphic (is that right?), and group with order 8 has 5 >nonisomorphic groups, Q8 and D8 are the 2 cases of non-abelian groups. >But they have the same character table, are they isomorphic?? Q8 is one >important group because of hamiltonian groups, which are non-abelian but >have only normal subgroups. of course, Q8 has only normal subgroups. But >it seems D8 has only normal subgroups too. Therefore, all hamiltonian >groups are consist of Q8*A*B is not right ?!? Here A, B are two special >abelian groups. > >I hope someone can give me some explainations of above confusion. >Thank you. >Bao > > >Sent via Deja.com http://www.deja.com/ >Before you buy. -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.