Assignment-1 (MTH-610)
Group Theory
Normal subgroups, Group homomorphisms
(1) Show that if G is a group of even order then there are exactly an odd
number of elements of order 2.
(2) Let p be a prime number. Prove Wilson’s theorem,
(p − 1)! ≡ −1(mod p)
(3) Let p be a fixed prime number and υ : Q× → Z be a map defined as
m
υ( ) = r
n
m
where r is the maximum power of p appearing in m
n i.e. to say n =
0
0
0
pr m
n0 , (p, m ) = 1, (p, n ) = 1. Is υ a group homomorphism?
If we modify the above map to
υ :Q→Z∪∞
by defining υ(0) = ∞ then will it be a homomorphism ?
(4) Let G be a group of order 12. Show that either G has a normal subgroup
of order 3 or G is isomorphic to A4 .
Solvable groups, Jordan Hölder theorem
(1) Find a composition series for A4 . Deduce that it is solvable.
(2) Show that a group G is solvable if and only if G(n) is the trivial group
for some n ≥ 0. Note that G(1) = [G, G], G(i) = [G(i−1) , G(i−1) ], i > 1.
(3) Show that The Jordan-Hölder theorem implies the fundamental theorem
of arithmetic.
(4) Show that the group of all upper triangular matrices of the form


1 a b
 0 1 c 
0 0 1
where a, b, c ∈ R is solvable.
Free Groups/Fintely presented groups
(1) Describe the universal property of free groups.
(2) Show that SL2 (Z) is a generated by the following two matrices,
0 −1
1 1
1 0
0 1
Is SL2 (Z) a free group on these generators?
(3) Prove that every group has a presentation.
(4) Find the generators and relations for GL2 (F2 ).
1
2
Group Actions
(1)
(2)
(3)
(4)
(5)
Show that all cycles of the same length in Sn are conjugates.
Describe the class equation for A4 .
If G is a non-trivial p-group then show that it has non-trivial center.
Give examples of faithful, transitive and free group actions.
Let H and K be two subgroups of G. Then show that the action of G
by left multiplication on the coset spaces G/H and G/K are equivalent
if H and K are conjugate subgroups.
Sylow’s theorem
Let p be a prime number.
(1) Classify all groups of order 15.
(2) Let G := GLn (Fp ). What is the order of a Sylow-p subgroup of G?
How many Sylow p subgroups are in G? Give one example of a Slow p
subgroup in G.
(3) Show that any group of order 45 is abelian.