By S. B. Kizlik

ISBN-10: 1572229225

ISBN-13: 9781572229228

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**Sample text**

For a given cycle a, the index of Nc(a) in Sn is therefore (n -1)1. Since ISnl = n! it follows that INc(a)1 = n. Since a has n distinct powers which commute with it, no other elements of Sn can commute with a. : 3. Cycles of length n are even permutations, so are in An. Suppose that a is a cycle of length n. /n = Hn - I)! elements. Since there are (n - 1)1 cycles of length n, there must be two conjugacy classes each containing ~(n - 1)1 elements. Then n - 1 is odd, so a cycle of length n - 1 is even and therefore belongs to An.

Let x have order m and y have order n. Then we have that (xy)mn = 1, and so xy has order at most mn. Suppose now that zmn = 1 where m and n are coprime. Then there are integers a and b such that am + bn = 1, wheilce z = zhn zam = xy where x == zhn and y = zam. It follows that x m == z',mn = 1 and yn = zamn == 1. The orders of x and y are m and n respectively, for if their orders were less, say m' and n', then z would be of order at most m' n', a contradiction. 45 Book 5 Groups A similar argument extends this result to the case where z is of order ml m2 ...

M). Then 6 8 =(1 8+1 28+1 0(2 (k-1)8+1) 8 + 2 28 + 2 .. (k - 1)8 -1- 2) o ... m o (k m 8+ k m 28 + k m (k-1)s+-) k where k is the least positive integer such that ks is divisible by m. L( m, s). ( m, 8) cycles in the decomposition as required. 18 We can choose a in p - 1 ways since 7l. p contains p - 1 non-zero elements. e. d = a-I (1 + be)). Thus there are p2(p _ 1) elements of this form. Consider now [~ ~]. As d is arbitrary it can be chosen in p ways. Since -be = 1 the element b must be non-zero, so can be chosen in 39 Book 5 Groups p - 1 ways.

### Algebra, part 2 (tablets) by S. B. Kizlik

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