数学代写 - Mathematics 322 — Term Test
时间:2020-11-20
Question 1.[4 marks] Determine whether or not the permutations σ = (1234)(256)(23) and
(3412)(134)(16) are conjugate in S6.
Question 2. Consider the group G = S6. For each i, let Hi be the subgroup of G fixing the
element i. Let X = {H1, . . . , H6}.
a) [3 marks] Show that if 1 ≤ i ≤ 6 and g ∈ G, then gHig 1 = Hj
for some j depending
on i and g, and that the function φ(g) : Hi 7→ Hj
is a permutation of the set X.
b) [2 marks] Show that g 7→ φ(g) is a homomorphism G → SX, where SX is the group of
permutations of X.
c) [6 marks] Show that the action defined by the homomorphism φ in part (b) is both
transitive and faithful. For each Hi
, find the stabilizer in G of Hi.