# Modern

## Basics, Groups and Subgroups¶

### ¶

Let \(m_{1}, m_{2}, \cdots, m_{k}\) be positive integers and \(d>0\) the greatest common divisor of \(m_{1}, m_{2}, \cdots, m_{k}\). Show that there exist integers \(x_{1}, x_{2}, \cdots, x_{k}\) such that

### 1b 2013¶

Give an example of an infinite group in which every element has finite order.

### 1a 2012¶

- (a) 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 \(b a b^{-1}=a^{-1}\).

### 1a 2011¶

- (a) Show that the set

is a non-abelian group of order 6 w.r.t. composition of mappings. \(\&\)

### 4a 2011¶

- (a) Let a and b be elements of a group, with \(\mathrm{a}^{2}=\mathrm{e}\), \(b^{6}=e\) 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 \(b^{m} a^{n}\). 20

### 1a 2010¶

(a) Let \(G=\mathbb{R}-\{-1\}\) 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 12

### 1a 2020 IFoS¶

Let \(p\) be a prime number. Then show that

Also, find the remainder when \(6^{44} \cdot(22) !+3\) is divided by 23 .

### 3a 2019 IFoS¶

If in the group \(G, a^{5}=e, a b a^{-1}=b^{2}\) for some \(a, b \in G\), find the order of \(b\).

### 1a 2017 IFoS¶

- (a) Prove that every group of order four is Abelian.

### 2a 2017 IFoS¶

- (a) Let \(G\) be the set of all real numbers except -1 and define \(a * b=a+b+a b\) \(\forall a, b \in G\). Examine if \(G\) is an Abelian group under *.

### 1a 2016 IFoS¶

1.(a) Prove that the set of all bijective functions from a non-empty set \(X\) onto itself is a group with respect to usual composition of functions.

### 1a 2014 IFoS¶

(a) If \(G\) is a group in which \((a \cdot b)^{4}=a^{4} \cdot b^{4},(a \cdot b)^{5}=a^{5} \cdot b^{5}\) and \((a \cdot b)^{6}=a^{6} \cdot b^{6}\), for all \(a, b \in G\), then prove that \(G\) is Abelian.

### 1b 2013 IFoS¶

(b) Prove that if every element of a group \((G, 0)\) be its own inverse, then it is an abelian group.

### 1a 2011 IFoS¶

(a) Let \(G\) be a group, and \(x\) and \(y\) be any two elements of \(G\). If \(y^{5}=e\) and \(y x y^{-1}=x^{2}\), then show that \(O(x)=31\), where \(e\) is the identity element of \(G\) and \(x \neq e\).

### 1a 2010 IFoS¶

(a) Let

Show that ' \(G\) is a group under matrix multiplication.

### 3a 2010 IFoS¶

(a) Show that zero and unity are only idempotents of \(Z_{n}\) if \(n=p^{r}\), where \(p\) is a prime.

## Cyclic Groups¶

2 a 2020

Let \(G\) be a finite cyclic group of order \(n\). Then prove that \(G\) has \(\phi(n)\) generators (where \(\phi\) is Euler's \(\phi\)-function).

### 2b 2016¶

Let \(p\) be a prime number and \(\mathbf{Z}_{\mathrm{p}}\) denote the additive group of integers modulo p. Show that every non-zero element of \(\mathbf{Z}_{p}\) generates \(\mathbf{Z}_{\mathbf{p}}\).

### 1a 2015¶

How many generators are there of the cyclic group \(G\) of order 8 Explain.

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.

### 1e 2011¶

(e) (i) Prove that a group of Prime order is abelian.

(ii) How many generators are there of the cyclic group \((\mathrm{G}, \cdot)\) of order 8 ?

### 2a 2011¶

- (a) Give an example of a group \(\mathrm{G}\) in which every proper subgroup is cyclic but the group itself is not cyclic.

### 1b 2010¶

(b) 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.

### 1b 2009¶

(b) Determine the number of homomorphisms from the additive group \(\mathbb{Z}_{15}\) to the additive group \(\mathbb{Z}_{10}\). ( \(\mathbb{Z}_{n}\) is the cyclic group of order \(n\) ). \(\quad 12\)

### 3b 2020 IFoS¶

Let \(G\) be a finite group and let \(p\) be a prime. If \(p^{m}\) divides order of \(G\), then show that \(G\) has a subgroup of order \(p^{m}\), where \(m\) is a positive integer.

### 2b 2016 IFoS¶

2.(b) Let \(\bar{G}\) be a group of order \(p q\), where \(p\) and \(q\) are prime numbers such that \(p>q\) and \(q \nmid(p-1)\). Then prove that \(G\) is cyclic.

### 1a 2015 IFoS¶

Q1. (a) If in a group \(G\) there is an element \(a\) of order 360 , what is the order of \(a^{220}\) ? Show that if \(G\) is a cyclic group of order \(\mathrm{n}\) and mivides \(\mathrm{n}\), then \(G\) has a subgroup of order \(m\).

### 3b 2011 IFoS¶

(b) Let \(G\) be a group of order \(2 p, p\) prime. Show that either \(G\) is cyclic or \(G\) is generated by \(\{a, b\}\) with relations \(a^{p}=e=b^{2}\) and \(b a b=a^{-1}\).

## Normal Subgroups¶

## Homomorphism¶

### 2a 2019¶

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.

### 2a 2018¶

Show that the quotient group of \((\mathbb{R},+)\) modulo \(\mathbb{Z}\) is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Z}\) is the set of integers.

### 2a 2019¶

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.

### 2a 2018¶

Show that the quotient group of \((\mathbb{R},+)\) modulo \(\mathbb{Z}\) is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Z}\) is the set of integers.

### 3a 2017¶

Show that the groups \(\mathbb{Z}_{5} \times \mathbb{Z}_{7}\) and \(\mathbb{Z}_{35}\) are isomorphic.

### 2a 2010¶

Let \(\left(\mathbb{R}^{*}\right.\), \()\) be the multiplicative group of nonzero reals and \((G L(n, \mathbb{R}), X)\) be the multiplicative group of \(n \times n\) non-singular' real matrices. Show that the quotient group \(G L(n, \mathbb{R}) / S L(n, \mathbb{R})\) and \(\left(\mathbb{R}^{*}, \cdot\right)\) are isomorphic where

\(S L(n, \mathbb{I})=\{A \in G L(n, \mathbb{I R}) / \operatorname{det} A=1\}\).

What is the centre of \(G L(n\), IR) ?

### 1a 2009¶

If \(\mathbb{R}\) is the set of real numbers and \(\mathbb{R}_{+}\)is the set of positive real numbers, show that \(\mathbb{R}\) under addition \((\mathbb{R},+)\) and \(\mathbb{R}\), under multiplication \(\left(\mathbb{R}_{+}, \cdot\right)\) are isomorphic. Similarly if \(\mathbb{Q}\) is the set of rational numbers and \(Q_{+}\)the set of positive rational numbers, are \((\mathbb{Q},+)\) and \(\left(\mathbb{Q}_{+}, \cdot\right)\) isomorphric? Justify your answer. \(4+8=12\)

### 2a 2018 IFoS¶

Find all the homomorphisms from the group \((\mathbb{Z},+)\) to \(\left(\mathbb{Z}_{4},+\right)\).

### 2a 2011 IFoS¶

Let \(G\) be the group of non-zero complex numbers under multiplication, and let \(N\) be the set of complex numbers of absolute value 1 . Show that \(G / N\) is isomorphic to the group of all positive real numbers under multiplication.

### 2b 2010 IFoS¶

Prove or disprove that \((\mathbb{R},+)\) and \(\left(\mathbb{R}^{+}, \cdot\right)\) are isomorphic groups where \(\mathbb{R}^{+}\) denotes the set of all positive real numbers.