Write out the conjugacy classes explicitly

Question

Questions:

Conjugacy classes

(a) Write out the conjugacy classes explicitly in S₃ and S₄.

(b) What are the conjugacy classes in A₄?

(c) Since |A₄|=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.