Problems 1A 7

1A.5. An action of a group G on a set fi is transitive if fi consists of a

single orbit. Equivalently, G is transitive on 0 if for every choice of points

a,/? E f2, there exists an element g E G such that a*g — j3. Now assume

that a group G acts transitively on each of two sets Vt and A. Prove that

the natural induced action of G on the cartesian product Q x A is transitive

if and only if GaGp — G for some choice of a E Vt and j3 E A.

Hint. Show that if GaGp = G for some a E ft and (3 E A, then in fact, this

holds for all a E fi and /? E A.

1A.6. Let G act on fi, where both G and fi are finite. For each element

g E G, write x(s) = |{a G fi | a-g = a}|. The nonnegative-integer-valued

function \

ls

called the permutation character associated with the action.

Show that

J(?) = ^ |Ga| = n|G|,

geG aGO

where n is the number of orbits of G on fl.

Note. Thus the number of orbits is

n

= T7^E^)'\G\geG

which is the average value of \

o v e r

the group. Although this orbit-counting

formula is often attributed to W. Burnside, it should (according to P. Neu-

mann) more properly be credited to Cauchy and Frobenius.

1A.7. Let G be a finite group, and suppose that H G is a proper sub-

group. Show that the number of elements of G that do not lie in any

conjugate of H is at least \H\.

Hint. Let % be the permutation character associated with the right-multipli-

cation action of G on the right cosets of H. Then Ylx{9) — \G\, where the

sum runs over geG. Show that ]T)x(M 2|if|, where here, the sum

runs over h E H. Use this information to get an estimate on the number of

elements of G where \ vanishes.

1A.8. Let G be a finite group, let n 0 be an integer, and let C be

the additive group of the integers modulo n. Let Q be the set of n-tuples

(#i, ^2,... , xn) of elements of G such that ^1^2 • • • xn = 1.

(a) Show that C acts on Q according to the formula

(#1, X2, • • • , Xn)*k = (Xi+fc, X2+k, • • • , Xn+k) ,

where k E C and the subscripts are interpreted modulo n.