By Larry C. Grove

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**Example text**

Suppose K is a conjugacy class in S , of cycle type ( k , , . . ,k,,), and that 25. If K A,,. If o E K write L for the conjugacy class of a in A,,. (1) If either k , , > 0 or k2,,,+ > 1 for some rn show that L = K . (2) If k , , = 0 and kz,+ I I 1 for all rn show that K = L u L’, where L’ is also a conjugacy class in A , and )L’I = JLI = IKl/2. G 27. Show that the group of rotational symmetries of the tetrahedron is isomorphic with A,. 28. Show that the group of rotational symmetries of the cube is isomorphic with S,.

Or E~ = glE Let ( U , ,E and ~ be ) 23 another universal pair for G. We have E = g2El. Thus E~ = g1g2s, and E = g2glE. But then we have and by the uniqueness in the definition of a universal pair we see that g2gl = l,,, the identity map on U . Similarly glg2 is the identity map on U , , and so g1 and g 2 are inverse isomorphisms. 1. If ( U , E ) is a universal pair for a group G and h E Aut(U) show that ( U , he) is also universal for G. Conversely if (U, E ~ is) universal for G show that = h~ for some h E Aut(U).

The kernel off is the normal subgroup of S,, consisting of all even permutations; it is called the alternating group on n letters and is denoted by A,,. 2. The alternating group A , has index 2 in the symmetric group S,, if n 2 2. Proof (Spitznagel [35]). If we show that the homomorphism f above maps S, onto H, then &/A, z H by the FHT, so IS,/A,,I = [$,:A,] = IHI = 2. Thus we need only prove the existence of an odd permutation. If the transposition (12) were even we would be able to write the identity permutation 1 as a product of an odd number of transpositions.