Search results for group+theory / GATE Overflow for GATE CSE

33 votes

6 answers

1

GATE CSE 1996 | Question: 1.4
Which of the following statements is FALSE? The set of rational numbers is an abelian group under addition The set of integers in an abelian group under addition The set of rational numbers form an abelian group under multiplication The set of real numbers excluding zero is an abelian group under multiplication

Which of the following statements is FALSE?The set of rational numbers is an abelian group under additionThe set of integers in an abelian group under additionThe set of ...

Kathleen

23.0k views

Kathleen asked Oct 9, 2014

77 votes

6 answers

2

GATE CSE 2014 Set 3 | Question: 50
There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$'s and $y$'s in some order. It is known that $x*x=y*y=x*y*x*y=y*x*y*x=e$ where $e$ is the identity element. The maximum number of elements in such a group is ____.

There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$'s and $y$'s in some order. It is ...

go_editor

15.6k views

go_editor asked Sep 28, 2014

37 votes

9 answers

3

GATE CSE 2019 | Question: 10
Let $G$ be an arbitrary group. Consider the following relations on $G$: $R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a = g^{-1}bg$ ... $R_1$ and $R_2$ $R_1$ only $R_2$ only Neither $R_1$ nor $R_2$

Let $G$ be an arbitrary group. Consider the following relations on $G$:$R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a ...

Arjun

17.3k views

Arjun asked Feb 7, 2019

60 votes

5 answers

4

GATE CSE 2007 | Question: 21
How many different non-isomorphic Abelian groups of order $4$ are there? $2$ $3$ $4$ $5$

How many different non-isomorphic Abelian groups of order $4$ are there?$2$$3$$4$$5$

Kathleen

19.6k views

Kathleen asked Sep 21, 2014

2 votes

2 answers

5

GATE CSE 2024 | Set 2 | Question: 53
Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \times Z_{4}$ that are their own inverses is ___________.

Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \ti...

Arjun

2.0k views

Arjun asked Feb 16

43 votes

5 answers

6

GATE CSE 2006 | Question: 3
The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false? It is not closed $2$ does not have an inverse $3$ does not have an inverse $8$ does not have an inverse

The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false?It is not closed$2$ does no...

Rucha Shelke

9.9k views

Rucha Shelke asked Sep 16, 2014

10 votes

1 answer

7

GATE CSE 2023 | Question: 41
Let $X$ be a set and $2^{X}$ denote the powerset of $X$. Define a binary operation $\Delta$ on $2^{X}$ as follows: \[ A \Delta B=(A-B) \cup(B-A) \text {. } \] Let $H=\left(2^{X}, \Delta\right)$. Which of the following statements about $H$ is/are correct? ... $A \in 2^{X},$ the inverse of $A$ is the complement of $A$. For every $A \in 2^{X},$ the inverse of $A$ is $A$.

Let $X$ be a set and $2^{X}$ denote the powerset of $X$.Define a binary operation $\Delta$ on $2^{X}$ as follows:\[A \Delta B=(A-B) \cup(B-A) \text {. }\]Let $H=\left(2^{...

admin

5.6k views

admin asked Feb 15, 2023

27 votes

4 answers

8

GATE CSE 2010 | Question: 4
Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ forms A Group A Ring An integral domain A field

Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ formsA GroupA R...

gatecse

9.8k views

gatecse asked Sep 21, 2014

41 votes

3 answers

9

GATE CSE 1994 | Question: 1.10
Some group $(G, o)$ is known to be abelian. Then, which one of the following is true for $G$? $g=g^{-1} \text{ for every } g \in G$ $g=g^2 \text{ for every }g \in G$ $(goh)^2 = g^2oh^2 \text{ for every } g, h \in G$ $G$ is of finite order

Some group $(G, o)$ is known to be abelian. Then, which one of the following is true for $G$?$g=g^{-1} \text{ for every } g \in G$$g=g^2 \text{ for every }g \in G$$(goh)...

Kathleen

10.6k views

Kathleen asked Oct 4, 2014

27 votes

4 answers

10

GATE CSE 1995 | Question: 2.17
Let $A$ be the set of all non-singular matrices over real number and let $*$ be the matrix multiplication operation. Then $A$ is closed under $*$ but $\langle A, *\rangle$ is not a semigroup. $\langle A, *\rangle$ is a semigroup but not a monoid. $\langle A, * \rangle$ is a monoid but not a group. $\langle A, *\rangle$ is a a group but not an abelian group.

Let $A$ be the set of all non-singular matrices over real number and let $*$ be the matrix multiplication operation. Then$A$ is closed under $*$ but $\langle A, *\rangle$...

Kathleen

9.9k views

Kathleen asked Oct 8, 2014

45 votes

4 answers

11

GATE CSE 1996 | Question: 2.4
Which one of the following is false? The set of all bijective functions on a finite set forms a group under function composition The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod $p$, where $p$ is a prime number The set of all strings over a finite ... $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$

Which one of the following is false?The set of all bijective functions on a finite set forms a group under function compositionThe set $\{1, 2, \dots p-1\}$ forms a group...

Kathleen

9.6k views

Kathleen asked Oct 9, 2014

3 votes

1 answer

12

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 28
A group $G$ in which $(a b)^2=a^2 b^2$ for all $a, b$ in $G$ is necessarily finite cyclic abelian none of the above

A group $G$ in which $(a b)^2=a^2 b^2$ for all $a, b$ in $G$ is necessarilyfinitecyclicabeliannone of the above

GO Classes

355 views

GO Classes asked Jan 28

2 votes

1 answer

13

GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 19
Let $\ast $ be the binary operation on the rational numbers given by $a \ast b=a+b+2 a b$. Which of the following are true? $\ast $ is commutative There is a rational number that is a $\ast \;-$ identity. Every rational number has a $\ast \;-$ inverse. I only I and II only I and III only I, II, and III

Let $\ast $ be the binary operation on the rational numbers given by $a \ast b=a+b+2 a b$. Which of the following are true?$\ast $ is commutativeThere is a rational numbe...

GO Classes

461 views

GO Classes asked Jan 21

3 votes

0 answers

14

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 61
Let $S$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Consider the two binary operations + and $\circ$ on $S$ ... law $(g+h) \circ f=(g \circ f)+(h \circ f)$. None III only II and III only I, II, and III

Let $S$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Consider the two binary operations + and $\circ$ on $S$ defined as pointwise addition and comp...

GO Classes

431 views

GO Classes asked Jan 28

2 votes

1 answer

15

GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 37
If $b$ and $c$ are elements in a group $G$, and if $b^5=c^3=e$, where $e$ is the unit element of $G$, then the inverse of $b^2 c b^4 c^2$ must be $b^4 c^2 b^2 c$ $c^2 b^4 c b^2$ $c b^2 c^2 b^4$ $c b c^2 b^3$

If $b$ and $c$ are elements in a group $G$, and if $b^5=c^3=e$, where $e$ is the unit element of $G$, then the inverse of $b^2 c b^4 c^2$ must be$b^4 c^2 b^2 c$$c^2 b^4 c...

GO Classes

427 views

GO Classes asked Jan 13

40 votes

9 answers

16

GATE CSE 2009 | Question: 22
For the composition table of a cyclic group shown below: ... $a,b$ are generators $b,c$ are generators $c,d$ are generators $d,a$ are generators

For the composition table of a cyclic group shown below:$$\begin{array}{|c|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c} & \textbf{d}\\\hline \textbf{a...

gatecse

8.9k views

gatecse asked Sep 15, 2014

14 votes

2 answers

17

GATE CSE 2022 | Question: 17
Which of the following statements is/are $\text{TRUE}$ for a group $\textit{G}?$ If for all $x,y \in \textit{G}, \; (xy)^{2} = x^{2} y^{2},$ then $\textit{G}$ is commutative. If for all $x \in \textit{G}, \; x^{2} = 1,$ then ... $2,$ then $\textit{G}$ is commutative. If $\textit{G}$ is commutative, then a subgroup of $\textit{G}$ need not be commutative.

Which of the following statements is/are $\text{TRUE}$ for a group $\textit{G}?$If for all $x,y \in \textit{G}, \; (xy)^{2} = x^{2} y^{2},$ then $\textit{G}$ is commutati...

Arjun

7.6k views

Arjun asked Feb 15, 2022

41 votes

2 answers

18

GATE CSE 2014 Set 3 | Question: 3
Let $G$ be a group with $15$ elements. Let $L$ be a subgroup of $G$. It is known that $L \neq\ G$ and that the size of $L$ is at least $4$. The size of $L$ is __________.

Let $G$ be a group with $15$ elements. Let $L$ be a subgroup of $G$. It is known that $L \neq\ G$ and that the size of $L$ is at least $4$. The size of $L$ is __________....

go_editor

8.5k views

go_editor asked Sep 28, 2014

40 votes

4 answers

19

GATE CSE 2004 | Question: 72
The following is the incomplete operation table of a $4-$ ... last row of the table is $c\;a\;e\; b$ $c\; b\; a\; e$ $c\; b\; e\; a$ $c\; e\; a\; b$

The following is the incomplete operation table of a $4-$element group.$$\begin{array}{|l|l|l|l|l|} \hline \textbf{*} & \textbf{e}& \textbf{a} &\textbf{b} & \textbf{c}\\\...

Kathleen

6.8k views

Kathleen asked Sep 18, 2014

23 votes

4 answers

20

GATE CSE 2021 Set 1 | Question: 34
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct? Both $G$ and $H$ are always cyclic $G$ may not be cyclic, but $H$ is always cyclic $G$ is always cyclic, but $H$ may not be cyclic Both $G$ and $H$ may not be cyclic

Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct?Both $G$ and $H$ are always cyclic$G$ may...

Arjun

8.3k views

Arjun asked Feb 18, 2021

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