By Derek J. S. Robinson

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**Extra resources for A Course in the Theory of Groups (2nd Edition) (Graduate Texts in Mathematics, Volume 80)**

**Sample text**

E G;.. ) and (gA)-l (gil). It is an easy matter to check the validity of the group axioms. 4. ) such that g;, = 1;, for almost all A, that is, with finitely many exceptions, is called the external direct product, D = Dr G;,. ;'eA The G;, are the direct factors. Clearly D is a subgroup of C; in fact it is even a normal subgroup. In case A = {Al' A2 , ••• , An}, a finite set, we write D=G; "xG ; ' 2 x"·xG. ;'n Of course C = D in this case. Should the groups G;, be written additively, we shall speak of the direct sum of the G;" and write G;" $ G;'2 $ ...

The set of all n-homomorphisms from G to H is written Homn(G, H). 4. Thus 1m tX is an n-subgroup of G and Ker tX a normal n-subgroup of G. The isomorphism theorems for n-groups hold: here of course all homomorphisms are n-homomorphisms. " We can also speak of n-endomorphisms ( = n-homomorphisms from a group to itself) and n-automorphisms (= bijective nendomorphisms). These form sets End n G and Aut n G: clearly End n G ~ End G and Autn G :s; Aut G. The reader is urged to prove the theorems about n-groups just mentioned: in all cases the proofs are close copies of the original ones.

If Hand K are permutation groups on finite sets X and Y, show that the order of H K is IHI1Y1IKI. *2. Let G be a permutation group on a finite set X. If nEG, define Fix n to be the set of fixed points of n, that is, all x in X such that xn = X. Prove that the number of G-orbits equals _ 111 G L IFix(n)l. 1teG 3. Prove that a finite transitive permutation group of order > 1 contains an element with no fixed points. *4. If Hand K are finite groups, prove that the class number of H x K equals the product of the class numbers of Hand K.