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 \
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
= 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.
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