Tuts

1. (a) How many different ways can the cyclic group C3 of order three act on the (b) How many different ways can the cyclic group C4 of order four act on the [Consider orbit decompositions and apply the Orbit-Stabiliser Theorem.] 2. (a) Let i1, i2, . . . , ik be distinct points in Ω = {1, 2, . . . , n} and let σ be a permutation in Sn. By considering the effect on various points in Ω, or σ−1(i1 i2 . . . ik)σ = (i1σ i2σ . . . ikσ).
Deduce that two permutations in Sn are conjugate if and only if they have (b) Give a list of representatives for the conjugacy classes in S5. How many elements are there in each conjugacy class? Hence calculate the order ofand generators for the centralisers of these representatives.
3. Let G be a group and let Γ and ∆ be sets such that G acts on Γ and on ∆.
for all γ ∈ Γ, δ ∈ ∆ and x ∈ G. Verify that this is an action of G on the set Γ × ∆.
Verify that the stabiliser of the pair (γ, δ) in this action equals the intersectionof the stabilisers Gγ and Gδ (these being the stabilisers under the actions of G on Γ and ∆, respectively).
If G acts transitively on the non-empty set Ω, show that is an orbit of G on Ω × Ω. Deduce that G acts transitively on Ω × Ω if and only 4. (a) There is a natural action of Sn on Ω = {1, 2, . . . , n}. How many orbits (b) Repeat part (a) with the action of the alternating group An on Ω.
5. Let G be a group and H be a subgroup of G. Show that the normaliser NG(H) of H is the largest subgroup of G in which H is a normal subgroup.
[By largest, we mean that if L is any subgroup of G in which H is normal, thenL � NG(H). So you should check that (i) H � NG(H) and (ii) if H � L thenL � NG(H).] 6. Let G be a group and H be a subgroup of G. Let Ω be the set of right cosets of H in G. Define an action of G on Ω by (a) Verify that this action is well-defined and that it is indeed a group action.
(c) Show that the stabiliser of the coset Hx is the conjugate Hx of H.
(d) Let ρ: G → Sym(Ω) be the permutation representation associated to the (e) Show that ker ρ is the largest normal subgroup of G contained in H.
[That is, show that (i) it is a normal subgroup of G contained in H and(ii) if K is any normal subgroup of G contained in H then K � ker ρ.
This kernel is called the core of H in G and is denoted by CoreG(H).] 7. If H is a subgroup of G of index n, show that the index of the core of H in G 8. Let G be a group and let G act on itself by conjugation. Show that the kernel of the associated permutation representation ρ: G → Sym(G) is the centre Z(G) of G. Deduce that Z(G) is a normal subgroup of G.
9. Let G be a group and let Aut G denote the set of all automorphisms of G. Show that Aut G forms a group under composition.
For g ∈ G, let τg : G → G be the map given by conjugation by g; that is, Show that the map τ : g �→ τg is a homomorphism τ : G → Aut G. What is the kernel of τ?Write Inn G for the image of τ. Thus Inn G is the set of inner automorphismsof G. Show that Inn G is a normal subgroup of Aut G. [Hint: Calculate the effectof φ−1τgφ on an element x, where φ ∈ Aut G, τg ∈ Inn G and x ∈ G.]

Source: http://turnbull.mcs.st-and.ac.uk/~colva/topics/tut2.pdf