School of Mathematics and Statistics MT5824 Topics in Groups ...

CMRD 2010
School of Mathematics and Statistics
MT5824 Topics in Groups
Problem Sheet V: Direct and semidirect products
1. Give an example of two groups G and H and a subgroup of the direct product
G × H which does not have the form G 1 × H 1 where G 1 G and H 1 H.
2. Let M and N be normal subgroups of a group G. By considering the map
g → (Mg,Ng),
or otherwise, show that G/(M ∩ N) is isomorphic to a subgroup of the direct
product G/M × G/N.
3. Using Question 2, or otherwise, show that if m and n are coprime integers, then
C m × C n
∼ = Cmn .
4. Let X 1 , X 2 ,...,X n be non-abelian simple groups and let
G = X 1 × X 2 ×···×X n .
(In this question we will identify the concepts of internal and external direct
products. Thus we speak a subgroup of G containing a direct factor X i instead
of it containing the subgroup ¯X i in the notation of the lectures.)
Prove that a non-trivial normal subgroup of G necessarily contains one of the
direct factors X i . Hence show that every normal subgroup of G has the form
X i1 × X i2 ×···×X ik
for some subset {i 1 ,i 2 ,...,i k } of {1, 2,...,n}.
[Hint: If N is a non-trivial normal subgroup of G, choose a non-identity (x 1 ,x 2 ,...,x n ) ∈
N. Consider conjugating this element by (1,...,1,g,1,...,1).]
Now suppose that X 1 , X 2 ,...,X n are abelian simple groups. Is it still true that
every normal subgroup of the direct product has this form?
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5. Let p be a prime number.
(a) Show that Aut C p
∼ = Cp−1 .
(b) Show that Aut(C p × C p ) ∼ = GL 2 (F p )(whereF p = Z/pZ denotes the field of
p elements).
[For (a), let C p = x. Observe that an automorphism α is given by x → x m
where m is a representative for a non-zero element of F p . Recall the multiplicative
group of a finite field is cyclic.
For (b), write C p × C p additively and view it as a vector space over F p . Show
that automorphisms of the group then correspond to invertible linear maps.]
6. Let G = x be a cyclic group. Show that Aut G is abelian.
7. Show that the dihedral group D 2n is isomorphic to a semidirect product of a
cyclic group of order n by a cyclic group of order 2. What is the associated
homomorphism φ: C 2 → Aut C n ?
8. Show that the quaternion group Q 8 may not be decomposed (in a non-trivial
way) as a semidirect product.
[Hint: How many elements of order 2 does Q 8 contain?]
9. Show that the symmetric group S 4 of degree 4 is isomorphic to a semidirect
product of the Klein 4-group V 4 by the symmetric group S 3 of degree 3.
Show that S 4 is also isomorphic to a semidirect product of the alternating
group A 4 by a cyclic group of order 2.
10. Let G be a group of order pq, wherep and q are primes with p