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Conjugacy classes

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(a) Write out the conjugacy classes explicitly in S_3 and S_4.

(b) What are the conjugacy classes in A_4?

(c) Since |A_4|=12, any subgroup of order 6 would be normal. Use (b) to show that A_4 has no subgroup of 6. Conclude that the converse to LaGrange's theorem is false.

(d) Find a normal subgroup of order 4 in A_4.

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Solution Summary

The solution provides an example of working with conjugacy classes and subgroups.

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(a) The conjugacy classes of S_3 are

{()},
{(12), (23), (13)}
{(123), (132)}

Since S_3 has trivial centre (and there's a singleton conjugacy class corresponding to each element of the centre); also, conjugate elements have the same order.

Recall that, in S_n, any two permutations having the same cycle type (that is, containing the same number of
cycles in ...

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