## Problem Sheet V: Constructing Groups

### 1.A permutation group G ≤SXis called transitive if for all x, y ∈X there exists some σ ∈G such that xσ = y.

(a) Prove that every group is isomorphic to a transitive subgroup of some symmetric group.

(b) What is the smallest positive integer n such that a cyclic group C6 of order 6 is isomorphic to a transitive subgroup of Sn?

(c) What is the smallest positive integer n such that a cyclic group C6 of order 6 is isomorphic to a subgroup of Sn?

2.Let n be a positive integer. If σ is an arbitrary permutation in Sn, defifine  Sn+2by

Show that the map Ø: Sn Sn+2 by σØ =  is an injective homomorphism with im Ø An+2.

Deduce that every fifinite group is isomorphic to a subgroup of an alternating group.

### 3.Let F be a fifield and let V = Fnbe the vector space of row vectors with entries from F . Let  数学math代写

= {e1, e2,…, en} be the standard basis for V .

(a) If σ is a permutation in Sn, defifine a linear map Tσ : V →V by

eiTσ = eiσ      for i = 1, 2, . . . , n.

Show that TσTτ = Tστ for all σ, τ ∈ Sn.

(b) As we are writing maps on the right, the matrix of a linear transformation T with respect to is obtained by expressing eiT in terms of the basis and writing the coeffiffifficients along the rows of the matrix.

Let Aσ denote the matrix of Tσ with respect to . Describe the entries of Aσ. (That is, specify which entries equal 1 and which equal 0.)

(c) Show that the map Ø: Sn → GLn(F ) by σ Aσ is an injective homomorphism.

(d) Deduce that every fifinite group is isomorphic to a subgroup of GLn(F ) for some positive inte-ger n.

4.Let G and H be any groups.  数学math代写

(a) Show that G × H ≅ H × G.

(b) Show that G ≅ G × 1.

(c) Show that G × H is abelian if and only if both G and H are abelian.

(d) Show that if g ∈ G and h ∈ H, then

o ( (g, h) ) = lcm(o(g), o(h)).

[The left-hand side denotes the order of the element (g, h) in the direct product G × H.]

### 5.Construct three non-abelian groups of order 24 that are pairwise non-isomorphic. Prove that they are indeed not isomorphic to each other.  数学math代写

6.(a) Show that Q8 is directly indecomposable (that is, it cannot be written as Q8 ≅N for two non-trivial groups M and N).

(b) Show that a cyclic group Cpn of prime-power order is directly indecomposable. [Hint: How many subgroups of order p does it possess?]

7.Consider the direct product G = S3 × S3 of two copies of the symmetric group of degree 3.

(a) Let N = A3×A3. Find all normal subgroups of G that are contained in N. [Hint: First describe all the subgroups of N. What is the conjugate of an element (x, y) by an element (1, g)?]

(b) How many normal subgroups of G are there containing N?

(c) Find all the normal subgroups of G and hence all quotient groups of G.

### 8.Let n ∈N be odd, and consider the dihedral group D4n of order 4n, with the standard generators α and β. Let H = 〈αn〉and K = 〈α2, β〉. Prove that:  数学math代写

(a) both H and K are normal subgroups of D4n;

(b) H ≅ Z2and K ≅ D2n;

(c) H ∩ K = {( )} and HK = D4n;

and conclude that D4n Z2 × D2n.

9.Let G1, G2, …, Gk be groups and let D = G1 , G2 × ··· × Gk be their direct product.

(a) Show that D is indeed a group.

(b) Find a homomorphism from Gi with image i= { (1,…, 1, g, 1,…, 1) | g ∈ Gi }. Show that

i is a normal subgroup of D isomorphic to Gi.

(d) Show that D = 1 2 k.

(e) Show that

i ( 1 i-1 i+1 k) = 1 for all i.

10.Let G be a group possessing normal subgroups N1, N2, …, Nk such that G = N1N2 …Nk and

Ni (N1 …Ni-1Ni+1 …Nk) = 1 for all i. Show that

G ≅ N1 × N2 × ··· × Nk.

11.Show that the converse of Proposition 5.12 is not true. That is, fifind groups G1 and G2 and a normal subgroup P G1 × G2 such that P M × N for any M  G1 and N  G2.