Mathematics 676 Homework (due Feb 3) A. Hulpke

Mathematics 676
Homework (due Feb 3)
A. Hulpke
1) For subgroups S, T ≤ G, a double coset is a subset of G of the form S gT = {sg t ∣ s ∈ S, t ∈ T}.
The set of all double cosets is denoted by S∖G/T . Show that the number of double cosets in S∖G/T
is equal to the number of orbits of T on the cosets of S.
2) Let G be a group The derived subgroup of G is G ′ = ⟨a −1 b −1 ab ∣ a, b ∈ G⟩. Show that if G = ⟨g⟩,
then G ′ = ⟨a−1 b −1 ab ∣ a, b ∈ g⟩ .
3) Let G = ⟨a = (1, 2, 3, . . . , 100), b = (1, 2)⟩ = S100 (you can assume this equality). We are computing the orbit of 1 under G. What representatives do we get?
Find a better generating set (with up to 3 generators), such that the words for representatives get
4) Show that any finite group G can be generated by log2 (∣G∣) elements.
Hint: A set of elements does not generate G, only if all elements lie in a proper subgroup.
5) a) Let U ≤ G with [G ∶ U] = n. Show that the probability that k randomly selected elements of
G are all contained in U is n1k .
b) Show that the probability that k random elements are simultaneously in any particular conjugate
of U (i.e. in a subgroup U g = g −1 Ug) is ≤ n k−1
c) Let m the number of conjugacy classes of maximal subgroups (i.e. subgroups U ≤ G such that
there is no U < V < G with proper inclusions) of G. (Usually m is small compared to ∣G∣.) Show
that, the probability of k random elements generating G is over 50%, if if 2k−1 > m.
(This is a justification for using only a few random Schreier generators.)
6) Suppose G = ⟨g⟩ is a cyclic group, acting on the domain Ω. How many nontrivial Schreier
generators will the Orbit/Stabilizer algorithm produce? (Why is this not surprising?)
7) Get accustomed with the system GAP .