Resent-Date: Sat, 15 Jul 95 19:19:28 BST From: Gotz PfeifferLarry Grove wrote Here are two groups: G = , H = . Both have order 605, they are not isomorphic. Can anyone find any "group-theoretical" (whatever that means) difference between the two? The structure of both groups clearly is 11^2 : 5, where the 5 acts as the matrix (ie. element of GL(2, 11)) [4 0] [4 0] C = [0 3] (mod 11) in G and Z = [0 5] (mod 11) in H. Note that det C = 4 * 3 = 12 = 1 (mod 11) and det Z = 4 * 5 = 20 = 9 (mod 11). Hence C lies in SL(2, 11) while Z doesn't. In other words: G is a subgroup of 11^2 : SL(2, 11), and H is but a subgroup of 11^2 : GL(2, 11). Two more remarks on G and H: as can be readily checked in GAP, the character tables of these groups are different. Their tables of marks, however, coincide. Best wishes, Goetz. ++++++++++++++++++++++++++++++++++ Resent-Date: Sun, 16 Jul 95 15:14:41 BST From: Werner Nickel As G"otz has pointed out, the action of the generators c and z can be represented by matrices C and Z in GL(2,11), respectively. The centralizer of C and the centralizer of Z in GL(2,11) is the subgroup of GL(2,11) consisting of the diagonal matrices. This is also the normalizer of in GL(2,11). The matrix [ 0 1 ] [ 1 0 ] normalizes the subgroup generated by C. This matrix together with the centralizer of C generates the normalizer of . A consequence of this is that the automorphism group of G is twice as large as the automorphism group of H. A straightforward computation with GAP shows that Aut(G) has order 24200 while Aut(H) has order 12100. Cheers, Werner. ++++++++++++++++++++++++++++++++++ Date: Mon, 17 Jul 1995 14:32:08 +1000 Geoff Smith Larry Grove asked for "group-theoretical" (whatever that means) differences between G = , and H = . -- and both G\"otz Pfieffer and Werner Nickel provided sensible answers. Of course there are many possible answers -- including the fatuous: For any group X consider the subgroup of X generated by all subgroups isomorphic to G etc. However, I suspect what Larry wants is a "natural" invariant" which discriminates between G and H and in that sense G\"otz's and Werner's postings are less silly. If you want to see some *really* similar non-isomorphic groups may I commend Eamonn O'Brien's list of 2-groups of order dividing 256. A few years ago Huseyin Aydin tried to discriminate (using natural invariants) between members of families of these groups which Eamonn O'Brien suggested might be very similar (see reference). There are some pairs of groups which are remarkably similar -- not only are the automorphism groups similar -- but even the power maps (x -> x^2) have the same shape. Geoff Smith Reference: H. AYDIN ``On Similar Distinct Finite 2-Groups'', Bath Mathematics and Computer Science Technical Report 90-42 (1990) +++++++++++++++++++++++++++++++ Resent-Date: Mon, 17 Jul 95 8:19:40 BST From: MANN@vms.huji.ac.il I want to draw the attention of this list to the paper `A Complete Set of Invariants for Finite Groups and Other Results', by M.Roitman, Adv. Math. 41 (1981), 301-311. Let n(X) be the number of subgroups of X. Then one of the results of this paper reads: Let G and H be finite groups such that n(L >< H) = n(L >< G), for each group L that is a subgroup of a quotient group of G. Then G and H are isomorphic. Avinoam Mann +++++++++++++++++++++++++++++++ Resent-Date: Mon, 17 Jul 95 12:05:07 BST From: "A.Camina" > Dear Pub-gang, > Here are two groups: > G = , > H = . > Both have order 605, they are not isomorphic. Can anyone find > any "group-theoretical" (whatever that means) difference between the > two? > Cheers, > Larry Grove > Doesn't this just say that the matrices (4,0;0,3) and (4,0;0,5) are not conjugate in GL(2,11)? Alan +++++++++++++++++++++++++++++++ Subject: not iso redux From: Larry Grove Resent-Date: Wed, 19 Jul 95 22:16:01 BST Many thanks for all the replies, both privately and via the pub-forum. As many noted the groups were cooked up via nonconjugate elements of order 5 in GL(2,11). I especially liked Werner's transition from there to the automorphism groups via normalizers. I'll append some observations made by Laci Kovacs (seemingly within minutes of the posting), as they differ somewhat from those already seen in replies to the forum. > The two minimal normal subgroups of one group are characteristic, >but those of the other are not. The automorphism group of one acts >nontrivially on the factor group of order 5, but that of other does >not. The multiplicator of one group is trivial, but that of the other >is not. Greetings, Laci. All the best to all, Larry G