By Harold Hilton
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Extra resources for An introduction to the theory of groups of finite order
Contain the inverse of each of their elements. 12 LEMMA. A o, AI' A 2 • As, A 6 • A IO ' and A I4 are the on~}' ambivalent alternating groups. 2 Conjugacy Classes of Symmetric and Alternating Groups be a product of disjoint cycles of odd lengths in cycle notation. We can form and call this permutation the standard conjugator. It satisfies (i) We notice first that ~ is even if and only if the number of cyclic factors of 1T whose lengths are congruent 3 modulo 4 is even. (ii) If the standard-conjugator is even for each element in a splitting class, then A n is ambivalent.
E. 2 12 or C s ( 7T) splits into exactly two An -classes of the same order 1C s ( 7T )1/2. e. if and only if Cs ( 7T) contains no odd permutations. Cs ( 7T) contains odd permutations if 7T contains a cyclic factor of even length (which is then an odd element of e 7T» or if 7T contains two cyclic factors of the same odd length, say s( so that Conversely. 1 Cs ( 7T) is generated by these cyclic factors, and hence is contained in An. 10 LEMMA. C S(7T) splits into two An-classes of equal order if and ollly if n> 1 and the nonzero parts of a( 7T ) are pairwise different and odd.
I+ I)(i+ I ..... 2i+ 1) so that (l, ... ,2i+ 1) is of the form pap-Ia-I where p: =(1. 1). Also. for i";;;j (l, .... 2i )(2i+ I, .... 2i+2j)= (1, ... , i+j+ I)(2i. i+j+ I, ... ,2i+2j), so this element is a commutator in S2i+2j' Similar remarks hold for arbitrary cycles of odd length and for each pair of disjoint cycles of even lengths. Thus each even permutation, since it consists of cycles with odd lengths together with an even number of cycles with even lengths (so that we can pair them off), is of form pap - la - 1 and is therefore a commutator.
An introduction to the theory of groups of finite order by Harold Hilton