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: $a, b \in G ⟹ a * b \in G$

2) Associative Law: $a, b, c \in G ⟹ a * (b * c) = (a * b) * c$

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 $| G |$ . 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:

$a, b \in H ⟹ a b \in H$
$a \in H ⟹ a^{- 1} \in H$

Cyclic Group

A group $G$ is called cyclic if $G = ⟨ g ⟩$ , where $⟨ g ⟩ = {g^{n} | n \in Z}$ 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 $g H g^{- 1} \subset H \forall g \in G$ . We write $H ⊴ G$ to denote that $H$ is a normal subgroup of $G$ .

If $G$ is abelian, then every subgroup is normal because $g H g^{- 1} = H g g^{- 1} = H$ .

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 $x H = {x h : h \in H}$ , and
the right coset of $H$ is the set $H X = {h x : h \in H}$

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 $g H = H g$ 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 $G = {f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}}$ of six transformations on the set of Complex numbers defined by $f_{1} (z) = z$ ,
$f_{2} (z) = 1 - z$ ,
$f_{3} (z) = \frac{z}{(1 - z)}$ ,
$f_{4} (z) = \frac{1}{z}$ ,
$f_{5} (z) = \frac{1}{(1 - z)}$ ,
$f_{6} (z) = \frac{(z - 1)}{z}$
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 $a * b = | a | b$ where $| a |$ denotes absolute value of $a$ . Does $(R_{0}, *)$ 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 $x^{2} \equiv - 1 (mod p)$ is solvable.

[2004, 12M]

12) Let $G$ be a group such that of all $a, b \in G$
i) $a b = b a$
ii) $g c d (O (a), O (b)) = 1$ ,
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]

Lagrange’s Theorem

TBC

Normal Subgroups

1) Let $G$ be the set of all real $2 \times 2 (\begin{array}{cc} x & y \\ 0 & z \end{array})$ , where $x z \neq 0$ matrices. Show that $G$ is group under matrix multiplication. Let $N$ denote the subset ${[\begin{array}{cc} 1 & a \\ 0 & 1 \end{array}] : a \in R}$ . 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 $a \neq 0$ 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 $| H | > \sqrt{| G |}$ and $| K | > \sqrt{G} |$ . 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 $\frac{G}{N}$ is abelian if and only if for all $x, y \in G$ , $x y x^{- 1} y^{- 1} \in N$ .

[2001, 20M]

Cosets and Quotient Groups

1) Write down all quotient groups of the group $Z_{12}$ .

[2019, 10M]

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