MTH204B - Problems
I.) Group, Subgroup, Cosets, Normal Subgroup, Homomorphisms
1. Let D be the set of 3 × 3 matrices with non-zero real entries along the diagonal and
with zeros above and below the diagonal. Show that it is a subgroup of GL3 (R)
w.r.t multiplication. Is it a normal subgroup?
0 i
0 1
2. i. Let Q be the subgroup of GL2 (C) generated by A =
and B =
.
i 0
−1 0
List the elements of Q.
ii. Check that Q4 := {±1, ±i, ±j, ±k} with multiplication defined by i2 = −1 = j 2
and ij = k = −ji is a group.
iii. Show that Q is isomorphic to Q4
3.
i. Let GL2 (Zp ) be the group of invertible 2 × 2 matrices with entries in the field
Zp , where p is a prime number. What is the order of GL2 (Zp )?
ii. Let H be the subgroup with matrices of determinant 1. What is the order of
H (Assume p ̸= 2).
4. If H is a non-empty finite subset of G which is closed w.r.t the group operation then
H is a subgroup of G.
5. Let Un denote the set of all i < n such that g.c.d(i, n) = 1.
i. Show that Un is a group with respect to multiplication modulo n.
ii. Show that U8 is not cyclic. Find its generators.
6. Every subgroup of an abelian group is normal. Give an example to show that the
converse is not true. Q8
a b
where ad = 1.
7. Let T2 (R) be the set of 2 × 2 real matrices of the form
0 d
i. Show that T2 (R) is a group with respect to multiplication.
1 b
ii. Show that U2 (R) = {
} is a normal subgroup of T2 (R).
0 1
iii. Show that T2 (R)/U2 (R) ∼
= R as groups.
8. Give an example of a group G and a subgroup H such that gHg −1 ⊂ H for some
g ∈ G but gHg −1 ̸= H.
9. Calculate D3 . Show that D3 ∼
= S3 .
10. Determine D4 . Is Q4 ∼
= D4 ?
11. Show that every group of order 4 is either Z/4Z or Z/2Z × Z/2Z
12. Let G1 × G2 be two groups and H1 and H2 be subgroups of G1 and G2 respectively.
i. Show that H1 × H2 is a subgroup of G1 × G2 .
ii. Is every subgroup of G1 × G2 of this form? In Z2*Z2,
G1 ×G2 ∼ G1
G2
iii. Show that H
.
= H1 × H
1 ×H2
2
H={(1,1),(0,0)} is not of the form H1*H2
13. (Chinese remainder theorem for groups)
i. Let m1 , . . . mr be r > 1 co-prime numbers. Show that the map Z → mZ1 Z × · · · ×
Z
mr Z given by n 7→ (n̄, . . . , n̄) is a surjective homomorphism and that its kernel
is m1 · · · mr Z.
1
ii. More generally, show that if H1 , . . . , Hr are r normal subgroups of a group G
with indices d1 , . . . dr which are co-prime then the map G → HG1 × · · · × HGr is
surjective with kernel H1 ∩ . . . ∩ Hr .
14. Let G1 and G2 be two groups of order m1 , m2 respectively such that (m1 , m2 ) = 1
then every subgroup of G is of the form H1 × H2 for some subgroup Hi in Gi , for
i = 1, 2.
15. Let G be a group and H ≤ G such that H ≤ Z(G) where Z(G) := {g ∈ G | xg =
gx∀x ∈ G}. Show that if G/H is cyclic then G is abelian.
16. Determine all the homomorphisms between (Q, +) → (Q, +); (Q/Z, +) → (Q, +).
17: Z(G1 × G2) = {(x,y) : x Z(G1), y Z(G2)} where Z(G1) and Z(G2) denote the centers of the groups G1 and G2, respectively.
17. Determine the center of G1 × G2 .
18. Let x be an element of G. If x belongs to H, then both xH and
Hx are equal to H. If x doesn't belong to H, then xH and Hx are
different cosets of H in G. But since the index of H in G is 2,
there are only two cosets of H in G, so xH must be equal to Hx.
Thus, for all x in G, xH = Hx. This means that H is normal in G
18. Show that a subgroup of index 2 is always normal.
II.) Dihedral group, Cyclic group
1. Show that the center of D2n+1 is trivial for n ∈ N. Determine the center of D2n .
2. Are D15 and D3 × Z5 isomorphic? 15 elements of order 2 in D6 but
3. Are D3 × Z2 and D6 isomorphic?
in D3*Z5 there is only 3 elements of order 2
yes define a mapping
4. Show that Z12 × Z14 is not cyclic. it doesnt have element of order 12*14
III.) (Free group) Let S be a set. Let F (S) be the set of finite words of the form xi11 · · · xinn
where n ∈ N and ij ∈ Z for all j = 1, . . . , n along with empty word ∅.
i. Show that F (S) is a monoid w.r.t the binary operation defined by concatenation
of words and the empty word ∅ is the identity under this operation.
ii. A subword of a word a1 . . . ak is a subexpression ai ai+1...ar for some 1 ≤ i ≤ r ≤ k.
A word in F (S) is reduced if it has no subword of the form aa−1 or a−1 a occurring
in it. With the convention that bb−1 = ∅ = b−1 b, show that by applying a finite
sequence of insertions of subwords aa−1 and deletions of subwords bb−1 , every word
in F (S) can be brought to a unique word in reduced form. (Refer: Rotman Ch. 5)
iii. Define an equivalence relation on F (S) given by u ∼ v if both have the same
reduced word. Denote by < S > the set of equivalence classes. Show that < S >
is a group with the induced group operation.
iv. Show that given a map ϕ : S → G, where G is a group there is a unique homomorphism ϕ̃ :< S >→ G such that ϕ̃|S = ϕ.
v. Show that every group can be realized as a quotient of a free group.
IV.) Order of elements, group actions, applications of counting principles
1. Let G be a group and a ∈ G have order k. If p is a prime divisor of k and there
exists a xp = a then order of x is kp.
2. Let G be a group of order 2p where p is an odd prime. If G has a normal subgroup
of order 2 then G is cyclic.
3. If G is a group such that there are atmost p elements that have order p then G is
cyclic. In particular, the multiplicative group F ∗ for a field F is always cyclic.
2
4. Let G be a group of order n and H is a subgroup of order m such that
G has proper normal subgroup.
n
m!
< 2n then
V.) Examples
1. An example of a non-cyclic group all of whose subgroups are cyclic. D4
2. An example of a non-abelian group all of whose subgroups are abelian. Q8
3. An example of an infinite group all of whose subgroups are finite.
4. An infinite group all of whose elements are of finite order.
5. A finitely generated group generated by elements of finite order but having elements
of infinite order. (Z,+)
6. Let m, n, r ∈ N. Give an example of a finite group G with elements a, b of order
m, n, resp., and ab is of order r. S3
7. Give an example where Q.I15 above fails if the condition ‘cyclic’ is replaced by
‘abelian’. G=Q8 and H= {1,-1,i,-i} so G/H is of order 2 so is cyclic but G is not cyclic even not abelian
4. circle group T, which is the group of complex numbers of unit magnitude under
multiplication. Every element in T has an order that is a divisor of the group order,
which is infinity.
15: Since G/H is cyclic, there exists an element gH in G/H such that the subgroup generated by gH is equal to G/H. This means that every element in G/H can be written
in the form (gH)^n for some integer n.
Let x, y be arbitrary elements of G. Then, we can write x = h_1 * g^m and y = h_2 * g^n for some h_1, h_2 in H and some integers m, n.
Since H is contained in the center of G, we have that hg = gh for all h in H and all g in G. Therefore, we have that xy = h_1 * g^m * h_2 * g^n = h_1 * h_2 * g^m * g^n.
Since H is contained in the center of G, it follows that h_1 * h_2 is also in H. Therefore, xy can be written in the form h * g^(m+n) for some h in H.
Similarly, we can write yx in the form h' * g^(m+n) for some h' in H. Since H is contained in the center of G, it follows that h * g^(m+n) = g^(m+n) * h for all h in H and
all g in G. Therefore, xy = yx, which implies that G is abelian.
Hence, if G/H is cyclic and H is contained in the center of G, then G is abelian.
3