We will cover following topics

Homomorphism Of Groups

Given two groups $(G, *)$ and $((H, \cdot))$ , a group homomorphism from $(G, *)$ to $((H, \cdot))$ is a function $h : G \to H$ such that for all $u$ and $v$ in $G$ , the following holds true:

h (u * v) = h (u) \cdot h (v)

where the group operation on LHS is that of $G$ and on RHS that of $H$ .

From the above definition, it can be deduced that $h$ maps the identity element $e_{G}$ of $G$ to the identity element $e_{H}$ of $H$ , that is,

h (e_{G}) = e_{H}

and $h$ also maps inverses to inverses, that is,

h (u^{- 1}) = h (u)^{- 1}

Therefore, $h$ is compatible with the group structure.

Image and Kernel

Let $h : G \to H$ be a homomorphism from $G$ to $H$ .

The kernel of homomorphism $h$ is defined to be the set of elements in $G$ which are mapped to the identity in $H$ ,

$\ker (h) = {u \in G : h (u) = e_{H}}$ ,

and the image of $h$ is defined as:

$im (h) = h (G) = {h (u) : u \in G}$

Isomorphism

A group homomorphism that is bijective (one-to-one and onto) is called isomorphism.

If such an isomorphism exists between two groups $G$ and $H$ , then the groups are said to be isomorphic to each other.

They differ only in the notation of their elements and are identical for all practical purposes.

First Isomorphism Theorem

Let $G$ and $H$ be groups and let $ϕ : G \to H$ be a homomorphism. Then,

(i) The kernel of $ϕ$ is a normal subgroup of $G$ ,

(ii) The image of $ϕ$ ia a subgroup of $H$ , and

(iii) The image of $ϕ$ is isomorphic to the quotient group $G / \ker (ϕ)$ .

Second Isomorphism Theorem

Let $G$ be a group, $S$ be a subgroup and $N$ be a normal subgroup of $G$ . Then, the following hold:

(i) The product $S N$ is a subgroup of $G$ ,

(ii) The intersection $S \cap N$ is a normal subgroup of $S$ , and

(iii) The quotient groups $(S N) / N$ and $S / (S \cap N)$ are isomorphic.

Third Isomorphism Theorem

Let $G$ be a group and $N$ a normal subgroup of $G$ . Then,

(i) If $K$ is a subgroup of $G$ such that $N \subseteq K \subseteq G$ , then $K / N$ is a subgroup of $G / N$ ,

(ii) Every subgroup of $G / N$ is of the form $K / N$ , for some subgroup $K$ of $G$ such that $N \subseteq K \subseteq G$ ,

(iii) If $K$ is a normal subgroup of $G$ such that $N \subseteq K \subseteq G$ , then $K / N$ is normal subgroup of $G / N$ ,

(iv) Every normal subgroup of $G / N$ is of the form $K / N$ , for some normal subgroup $K$ of $G$ such that $N \subseteq K \subseteq G$ , and

(v) If $K$ is a normal subgroup of $G$ such that $N \subseteq K \subseteq G$ , then the quotient group $(G / N) / (K / N)$ is isomophic to $G / K$ .

Sylow’s Group

If $p^{k}$ is the highest power of a prime $p$ dividing the order of a finite group $G$ , then a subgroup of $G$ of order $p^{k}$ is called a Sylow $p - s u b g r o u p$ of $G$ .

Permutation Groups

A permutation of a set $A$ is a one-one and onto function from $A$ to $A$ .

Let $A = {1, 2, \dots, n}$ be the set of first $n$ natural numbers.

The set of all permutations of $A$ is called the symmetric group of degree $n$ , and is denoted by $S_{n}$ .

The order of $S_{n}$ is $n!$ .

Cycle notation

The permutation $(\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 2 & 7 & 6 & 5 & 8 & 1 & 3 \end{matrix}) = (146837) (2) (5)$ , where the expression on RHS represents the cyclic permutation of the given permutation on RHS.

The missing elements in the cyclic notation are mapped to themselves.

For example, the permutation $(\begin{matrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 3 & 4 & 5 \end{matrix})$ can be written as $(1)$ or $(2)$ , etc. in the cyclic notation.

Cayley’s Theorem

According to Cayley’s theorem, every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$ .

PYQs

Homomorphism Of Groups

1) If $G$ and $H$ are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from $G$ to $H$ , the trivial one.

[10M]

2) If $R$ is a ring with unit element 1 and $ϕ$ is a homomorphism of $R$ onto $R^{'}$ , prove that $ϕ (1)$ is the unit element of $R^{'}$ .

[2015, 15M]

3) Determine the number of homomorphisms from the additive group $Z_{15}$ to the additive group $Z_{10}$ . ( $Z_{n}$ is the cyclic group of order $n$ ).

[2009, 12M]

4) Let $G$ and $\bar{G}$ be two groups and let $ϕ : G \to \bar{G}$ be a homomorphism. For any element $a \in G$ .
i) Prove that $O (ϕ (a)) / O (a)$

[2008, 15M]

ii) Ker $ϕ$ is normal subgroup of $G$

[2008, 15M]

5) Verify that the set $E$ of the four roots of $x^{4} - 1 = 0$ forms a multiplicative group. Also prove that a transformation $T$ , $T (n) = i^{n}$ is a homomorphism from $I_{+}$ (Group of all integers with addition) onto $E$ under multiplication.

[2004, 10M]

Isomorphism

1) Show that the quotiet group of $(R, +)$ modulo $Z$ is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex numbers on the unit circle in the complex plane. Here $R$ is the set of real numbers and $Z$ is the set of integers.

[2018, 15M]

2) Show that the groups $Z_{5} \times Z_{7}$ and $Z_{35}$ are isomorphic.

[2017, 15M]

3) Let $G$ be a group of order $n$ . Show that $G$ is isomorphic to a subgroup of the permutation group $S_{n}$ .

[2017, 10M]

4) Show that a cyclic group of order 6 is isomorphic to the product of a cyclic group of order 2 and a cyclic group of order 3. Can you generalize this? Justify.

[2010, 12M]

5) Let $(R^{*}, .)$ be the multiplicative group of non-zero reals and $(G L (n, R), X)$ be the multiplicative group of $n \times n$ non-singular real matrices. Show that the quotient group $\frac{G L (n, R)}{S L (n, R)}$ and $(R^{*}, .)$ are isomorphic where $S L (n, R) = {A \in G L (n, R) / \det A = 1}$ . What is the center of $G L (n, R)$ ?

[2010, 15M]

6) If $R$ is the set of real numbers and $R_{t}$ is the set of positive real numbers, show that $R$ under addition $(R, +)$ and $R_{+}$ under multiplication $(R_{+}, .)$ are isomorphic. Similarly if $Q$ is set of rational numbers and $Q_{+}$ is the set of positive rational numbers, are $(Q, +)$ and $(Q, .)$ isomorphic? Justify your answer.

[2009, 4+8=12M]

7) If $G$ is a group of real numbers under addition and $N$ is the subgroup of $G$ consisting of integers, prove that $\frac{G}{N}$ is isomorphic to the group $H$ of all complex numbers of absolute value 1 under multiplication.

[2006, 12M]

8) If $f : G \to G$ is an isomorphism, prove that the order $a \in G$ of is equal to the order of $f (a)$ .

[2005, 15M]

Sylow’s Groups

TBC

Permutation Groups

1) What are the orders of the following permutation in $S_{10}$ ?
$(\begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 1 & 8 & 7 & 3 & 10 & 5 & 4 & 2 & 6 & 9 \end{array})$ and $(12345) (67)$ .

[2013, 10M]

2) What is the maximal possible order of an element in $S_{10}$ ? Why? Give an example of such a element. How many elements will there be in $S_{10}$ of that order?

[2013, 13M]

3) How many conjugacy classes does the permutation group $S_{5}$ of permutation 5 numbers have? Write down one element in each class (preferably in terms of cycles).

[2012, 15M]

Cayley’s Theorem

TBC

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