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: \(a, b \in G \implies a * b \in G\)
2) Associative Law: \(a, b, c \in G \implies a *(b * c)=(a * b) * c\)
3) Existence of Identity Element e: There exists an element \(e \in G\) such that \(a * e=e * a=a \forall a \in G\)
4) Existence of Inverse in G: For every \(a \in G\), there exists an element \(a^{-1} \in G\) such that \(a * a^{-1}=a^{-1} * a=e\)
Abelian Group
A group \(G\) is called abelian (or commutative) if for every \(a, b \in G\), \(a*b=b*a\).
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 \(a, b, c \in G\), \(a *(b * c)=(a * b) * c\)
ii) Identity: There exists an element \(e \in G\) such that \(a * e=e * a=a \forall a \in G\)
iii) Inverse: For every \(a \in G\), there exists an element \(a^{-1} \in G\) such that \(a * a^{-1}=a^{-1} * a=e\)
iv) Closure: For all \(a, b \in G\), \(a * b \in G\)
v) Commutativity: For all \(a, b \in G\), \(a * b = b*a\)
Order of a Group
Let \((G,*)\) be a group and \(g \in G\).
The number of elements in \(G\) is called the order of the group and is denoted by \(\vert G \vert\). The order of \(g\) is the smallest positive integer \(n\) such that \(g^{n}=1\).
If there is no positive integer \(n\) such that \(g^{n}=1\), 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:
- \[a, b \in H \implies ab \in H\]
- \[a \in H \implies a^{-1} \in H\]
Cyclic Group
A group \(G\) is called cyclic if \(G=\langle g\rangle\), where \(\langle g\rangle=\left\{g^{n} \vert n \in \mathbb{Z}\right\}\) is the subgroup generated by \(g\).
If the generator \(g\) has order \(n\), \(G=\langle g\rangle\) 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 \(g H g^{-1} \subset H \forall g \in G\). We write \(H \trianglelefteq G\) to denote that \(H\) is a normal subgroup of \(G\).
If \(G\) is abelian, then every subgroup is normal because \(g H g^{-1}=H g g^{-1}=H\).
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 \(xH=\{x h : h \in H\}\), and
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the right coset of \(H\) is the set \(HX = \{h x : h \in H\}\)
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 \(G/H=\{g H, g \in G\}\) 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 \(b a b^{-1}=a^{-1}\).
[2012, 12M]
3) Show that the set \(G=\left\{f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}\right\}\) of six transformations on the set of Complex numbers
defined by \(f_{1}(z)=z\),
\(f_{2}(z)=1-z\),
\(f_{3}(z)=\dfrac{z}{(1-z)}\),
\(f_{4}(z)=\dfrac{1}{z}\),
\(f_{5}(z)=\dfrac{1}{(1-z)}\),
\(f_{6}(z)=\dfrac{(z-1)}{z}\)
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 \(a^{2}=e\), \(b^{6}=e\) and \(a b=b^{4} a\). Find the order of \(a b\), and express its inverse in each of the forms \(a^{m} b^{n}\) and \(b^{m} a^{n}\).
[2011, 20M]
6) Let \(G=R-\{-1\}\) be the set of all real numbers omitting -1. Define the binary relation \(*\) on \(G\) by \(a * b=a+b+a b\). Show \((G, *)\) is a group and it is abelian.
[2010, 12M]
7) Let \(R_{0}\) be the set of all real numbers except zero. Define a binary operation \(*\) on \(R_{0}\) as \(a * b= \vert a \vert b\) where \(\vert a \vert\) denotes absolute value of \(a\). Does \(\left(R_{0}, *\right)\) form a group? Examine.
[2008, 12M]
8) Prove that there exists no simple group of order 48.
[2007, 15M]
9) If in a group \(G\), \(a^{5}=e\), \(e\) is the identity element of \(G ~ a b a^{-1}=b^{2}\) for \(a, b \in G\), then find the order of \(b\).
[2007, 12M]
10) Let \(S\) be the set of all real numbers except -1. Define on \(S\) by \(a * b=a+b+a b\). 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 \(4 n+1\), \(n\) being a natural number, then show that congruence \(x^{2} \equiv-1(\bmod p)\) is solvable.
[2004, 12M]
12) Let \(G\) be a group such that of all \(a, b \in G\)
i) \(a b=b a\)
ii) \(gcd(O(a), O(b))=1\),
then show the \(O(a b)=O(a) O(b)\).
[2004, 12M]
13) Show that a group of \(p^{2}\) 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 \(K \subset H\). Show that \((G:K)=(G:H)(H:K)\).
[2019, 10M]
2) Find all the proper subgroups of the multiplicative group of the field \((Z_{13},+_{13},\times_{13})\), where \(+_{13}\) and \(\times_{13}\) represent addition modulo 13 and multiplication modulo 13 respectively.
[2020, 20M]
3) Let \(p\) be prime number and \(Z_{p}\) denote the additive group of integers modulo \(p\). Show that that every non-zero element \(Z_{p}\) generates \(Z_{p}\).
[2016, 15M]
4) Show that the alternating group of four letters \(A_{4}\) 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 \(2 \times 2 \left( \begin{array}{cc}{x} & {y} \\ {0} & {z}\end{array}\right)\), where \(x z \neq 0\) matrices. Show that \(G\) is group under matrix multiplication. Let \(N\) denote the subset \(\left\{\left[ \begin{array}{cc}{1} & {a} \\ {0} & {1}\end{array}\right]: a \in R\right\}\). 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 \(a \neq 0\) and define in \(G\) an operation as follows:
\((a, b) \otimes(c, d)=(a c, b c+d)\).
Examine whether \(G\) is a group wrt the operation \(\otimes\). If it is a group, is \(G\) abelian?
[2006, 10M]
3) If \(M\) and \(N\) are normal subgroups of a group \(G\) such that \(M \cap N=\{e\}\), 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 \(\vert H \vert >\sqrt{ \vert G \vert }\) and \(\vert K \vert>\sqrt{G} \vert\). Prove that \(H \cap K \neq\{e\}\).
[2005, 15M]
5) If \(H\) is a subgroup of a group \(G\) such that \(x^{2} \in H\) for every \(x \in G\), 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 \(\dfrac{G}{N}\) is abelian if and only if for all \(x, y \in G\), \(x y x^{-1} y^{-1} \in N\).
[2001, 20M]