Group Theory
We will cover following topics
Groups
An algebraic structure (G,∗), where G is a set of elements and ∗ is the binary operation defined on it, is called a group if it satisfies the following four properties:
1) Closure Property: a,b∈G⟹a∗b∈G
2) Associative Law: a,b,c∈G⟹a∗(b∗c)=(a∗b)∗c
3) Existence of Identity Element e: There exists an element e∈G such that a∗e=e∗a=a∀a∈G
4) Existence of Inverse in G: For every a∈G, there exists an element a−1∈G such that a∗a−1=a−1∗a=e
Abelian Group
A group G is called abelian (or commutative) if for every a,b∈G, a∗b=b∗a.
To qualify as an abelian group, the set and operation, (G,∗), must satisfy five requirements known as the abelian group axioms:
i) Associativity: For all a,b,c∈G, a∗(b∗c)=(a∗b)∗c
ii) Identity: There exists an element e∈G such that a∗e=e∗a=a∀a∈G
iii) Inverse: For every a∈G, there exists an element a−1∈G such that a∗a−1=a−1∗a=e
iv) Closure: For all a,b∈G, a∗b∈G
v) Commutativity: For all a,b∈G, a∗b=b∗a
Order of a Group
Let (G,∗) be a group and g∈G.
The number of elements in G is called the order of the group and is denoted by |G|. The order of g is the smallest positive integer n such that gn=1.
If there is no positive integer n such that gn=1, then g has infinite order.
Examples of Groups under addition operation
Some examples of groups under the addition operation are:
- R (the set of real numbers),
- Z (the set of integers),
- Q (the set of rational numbers) and
- C (the set of complex numbers
Note: From now onwards, we will write G to denote the algebraic structure (G,∗).
Subgroups
A subset H of a group G is called called a subgroup of G if H itself is a group under the operation on G.
We write H<G to denote that H is a subgroup of G.
Every group G has at least two subgroups- G itself and the subgroup e. All other subgroups are called proper subgroups.
Theorem: A non-empty subset H of the group G is a subgroup iff:
- a,b∈H⟹ab∈H
- a∈H⟹a−1∈H
Cyclic Group
A group G is called cyclic if G=⟨g⟩, where ⟨g⟩={gn|n∈Z} is the subgroup generated by g.
If the generator g has order n, G=⟨g⟩ is cyclic of order n.
Lagrange’s Theorem
It states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
Normal Subgroup
A subgroup H<G is called normal if gHg−1⊂H∀g∈G. We write H⊴G to denote that H is a normal subgroup of G.
If G is abelian, then every subgroup is normal because gHg−1=Hgg−1=H.
Cosets
Let H be a subgroup of G and let x be an element of G. Then,
-
the left coset of H is the set xH={xh:h∈H}, and
-
the right coset of H is the set HX={hx:h∈H}
Both the left and right cosets of H are subsets of G.
Quotient Group
Let G be a group and H be its subgroup such that gH=Hg for any g in G.
The set G/H={gH,g∈G} of cosets of H in G is called a quotient group.
PYQs
Groups
1) Give an example of an infinite group in which every element has finite order.
[2013, 10M]
2) How many elements of order 2 are there in the group of order 16 generated by a and b such that the order of a is 8, the order of b is 2 and bab−1=a−1.
[2012, 12M]
3) Show that the set G={f1,f2,f3,f4,f5,f6} of six transformations on the set of Complex numbers
defined by f1(z)=z,
f2(z)=1−z,
f3(z)=z(1−z),
f4(z)=1z,
f5(z)=1(1−z),
f6(z)=(z−1)z
is a non-abelian group of order 6 wrt composition of mappings.
[2011, 12M]
4) Prove that a group of Prime order is abelian.
[2011, 6M]
5) Let a and b be elements of a group, with a2=e, b6=e and ab=b4a. Find the order of ab, and express its inverse in each of the forms ambn and bman.
[2011, 20M]
6) Let G=R−{−1} be the set of all real numbers omitting -1. Define the binary relation ∗ on G by a∗b=a+b+ab. Show (G,∗) is a group and it is abelian.
[2010, 12M]
7) Let R0 be the set of all real numbers except zero. Define a binary operation ∗ on R0 as a∗b=|a|b where |a| denotes absolute value of a. Does (R0,∗) form a group? Examine.
[2008, 12M]
8) Prove that there exists no simple group of order 48.
[2007, 15M]
9) If in a group G, a5=e, e is the identity element of G aba−1=b2 for a,b∈G, then find the order of b.
[2007, 12M]
10) Let S be the set of all real numbers except -1. Define on S by a∗b=a+b+ab. Is (S,∗) a group? Find the solution of the equation 2∗x∗3=7 in S.
[2006, 12M]
11) If p is prime number of the form 4n+1, n being a natural number, then show that congruence x2≡−1(modp) is solvable.
[2004, 12M]
12) Let G be a group such that of all a,b∈G
i) ab=ba
ii) gcd(O(a),O(b))=1,
then show the O(ab)=O(a)O(b).
[2004, 12M]
13) Show that a group of p2 is abelian, where p is a prime number.
[2002, 10M]
Subgroups
1) Let G be a finite group, H and K subgroups of G such that K⊂H. Show that (G:K)=(G:H)(H:K).
[2019, 10M]
2) Find all the proper subgroups of the multiplicative group of the field (Z13,+13,×13), where +13 and ×13 represent addition modulo 13 and multiplication modulo 13 respectively.
[2020, 20M]
3) Let p be prime number and Zp denote the additive group of integers modulo p. Show that that every non-zero element Zp generates Zp.
[2016, 15M]
4) Show that the alternating group of four letters A4 has no subgroup of order 6.
[2009, 15M]
Cyclic Groups
1.(i) How many generators are there of the cyclic group G of order 8? Explain.
[2015, 5M]
1.(ii) Taking a group {e,a,b,c} of order 4, where e is the identity, construct composition tables showing that one is cyclic while the other is not.
[2015, 5M]
2) How many generators are there of the cyclic group (G,.) of order 8?
[2011, 6M]
3) Give an example of a group G in which every proper subgroup is cyclic but the group itself is not cyclic.
[2011, 15M]
4) Show that a group of order 35 is cyclic.
[2002, 12M]
Normal Subgroups
1) Let G be the set of all real 2×2(xy0z), where xz≠0 matrices. Show that G is group under matrix multiplication. Let N denote the subset {[1a01]:a∈R}. Is N a normal subgroup of G? Justify your answer.
[2014, 10M]
2.(i) Let O(G)=108. Show that there exists a normal subgroup or order 27 or 9.
[2006, 10M]
2.(ii) Let G be the set of all those ordered pairs (a,b) of real numbers for which a≠0 and define in G an operation as follows:
(a,b)⊗(c,d)=(ac,bc+d).
Examine whether G is a group wrt the operation ⊗. If it is a group, is G abelian?
[2006, 10M]
3) If M and N are normal subgroups of a group G such that M∩N={e}, show that every element of M commutes with every element of N.
[2005, 12M]
4) Let H and K be two subgroups of a finite group G such that |H|>√|G| and |K|>√G|. Prove that H∩K≠{e}.
[2005, 15M]
5) If H is a subgroup of a group G such that x2∈H for every x∈G, then prove that H is a normal subgroup of G.
[2003, 12M]
6) Prove that a group of order 42 has a normal subgroup of order 7.
[2002, 10M]
7) Let N be a normal subgroup of a group G. Show that GN is abelian if and only if for all x,y∈G, xyx−1y−1∈N.
[2001, 20M]