A First Course in Group Theory, 1st Edition by Cyril F. Gardiner (auth.)

By Cyril F. Gardiner (auth.)

One of the problems in an introductory ebook is to speak a feeling of objective. purely too simply to the newbie does the booklet develop into a chain of definitions, strategies, and effects which look little greater than curiousities top nowhere particularly. during this ebook i've got attempted to beat this challenge via making my valuable objective the decision of all attainable teams of orders 1 to fifteen, including a few learn in their constitution. by the point this objective is realised in the direction of the top of the e-book, the reader must have bought the elemental principles and strategies of staff idea. To make the publication extra beneficial to clients of arithmetic, specifically scholars of physics and chemistry, i've got incorporated a few functions of permutation teams and a dialogue of finite aspect teams. The latter are the best examples of teams of partic­ ular curiosity to scientists. They ensue as symmetry teams of actual configurations comparable to molecules. Many principles are mentioned generally within the workouts and the suggestions on the finish of the ebook. in spite of the fact that, such principles are used hardly ever within the physique of the ebook. after they are, compatible references are given. different workouts attempt and reinfol:'ce the textual content within the ordinary method. a last bankruptcy supplies a few proposal of the instructions within which the reader may fit after operating via this publication. References to aid during this are indexed after the description solutions.

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If S is any non-empty subset of the group G, then SG = G = GS. PROOF SG = {sg ISES, g (G}. S. Hence SG C G. Let g £ G and s (S, then g = s (S-lg) C SG. Thus G C SG. The 2 inclusions imply that G = SG. Similarly GS = G. Given the product AB, it is natural to ask if AB is a subgroup when both A and B are subgroups. The following theorem shows that AB is a subgroup only when certain conditions hold. 2. (1) If A and B are subgroups of a group G, then AB is itseLf a subgroup if and onLy if AB = BA When this is so, the group (A, B), generated by A and B, is equaZ to AB.

I n. 1 Therefore, by (2) above, n e. = mn I mn l . I nl . and n and n l are positive integers, we conclude that A similar argument shows that m Thus O(ab) = mn. Since (m. n) = 1, we have Mm + Nn (see [GJ) . g Nn Mm Nn Then g = ab = ba. Put a = g and b = g (6) 1 for suitable integers M and N Now since g has order mn. gm must have order n. Thus (gm)M has order n by (4) above, since from Mm + Nn = 1, we have (M. n) By a similar argument, gNn has order m. Thus =n =m O(a) and O(b) as required. O(a l ) To prove uniqueness, suppose we also had g = nand O(b l ) m.

Thus = AB. We have shown that :x:, y £ AB .... 1 (2) that AB is a subgroup. (2) (A, B):l AB because (A. S) contains all products like Now AB ;:) A and AB ;:) B. Thus AB :> (A, B). when AB is a subgroup. Hence, when AB is a subgroup, we have ab; a ' A. bE B. AB (A, B) n (3) Put D = A B. Now D is a subgroup of B so we can decompose B into right cosets relative to D as follows: , where the cosets are disjoint and b i E B; i 1, 2, ••• , l". 45 Then = AVb 1 tJ AB AVb 2 U ... Now D is a subgroup of A.

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