Recent questions tagged group-theory

2 votes

2 answers

1

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 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

asked Feb 18 in Set Theory & Algebra Arjun 347 views

0 votes

1 answer

2

NIELIT 2016 DEC Scientist B (IT) - Section B: 21
Consider the set $S=\{1,\omega,\omega ^2\}$, where $\omega$ and $\omega^2$ 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 $S=\{1,\omega,\omega ^2\}$, where $\omega$ and $\omega^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S,*)$ forms: A group A ring An integral domain A field

asked Mar 31, 2020 in Set Theory & Algebra Lakshman Patel RJIT 170 views

0 votes

1 answer

3

NIELIT 2016 DEC Scientist B (CS) - Section B: 28
Which one of the following is NOT necessarily a property of a Group? Commutativity Associativity Existence of inverse for every element Existence of identity

Which one of the following is NOT necessarily a property of a Group? Commutativity Associativity Existence of inverse for every element Existence of identity

asked Mar 31, 2020 in Set Theory & Algebra Lakshman Patel RJIT 144 views

1 vote

4 answers

4

NIELIT 2017 DEC Scientist B - Section B: 48
On a set $A = \{a,b,c,d\}$ a binary operation $*$ ... The relation is Commutative but not associative Neither commutative nor associative Both commutative and associative Associative but not commutative

On a set $A = \{a,b,c,d\}$ a binary operation $*$ is defined as given in the following table. $\begin{array}{|c|cccc|}\hline\text{*}&a&b&c&d\\\hline \text{a}&a&c&b&d\\\text{b}&c&b&d&a\\\text{c}&b&d&a&c\\\text{d}&d&a&c&b\\\hline \end{array}$ The relation is Commutative but not associative Neither commutative nor associative Both commutative and associative Associative but not commutative

asked Mar 30, 2020 in Set Theory & Algebra Lakshman Patel RJIT 632 views

8 votes

5 answers

5

GATE CSE 2020 | Question: 18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.

Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.

asked Feb 12, 2020 in Linear Algebra Arjun 3.5k views

0 votes

1 answer

6

$\textbf{NTA NET DEC 2019 (group)}$
Consider the following statements: $\mathbf{S_1:}\;\;$If a group $\mathbf{(G,*)}$ is of order $\mathbf n$ and $\mathrm {a \in G}$ is such that $\mathrm {a^m=e}$ for some integer $\mathrm {m \le n}$ then $\mathbf m$ must divide $\mathbf n$ ... $(3)\;\;\;\text{Niether}\; \mathrm{S_1}\;\text{nor}\;\mathrm{S_2}$

Consider the following statements: $\mathbf{S_1:}\;\;$If a group $\mathbf{(G,*)}$ is of order $\mathbf n$ and $\mathrm {a \in G}$ is such that $\mathrm {a^m=e}$ for some integer $\mathrm {m \le n}$ then $\mathbf m$ must divide $\mathbf n$. $\mathbf {S_2:}\;\;$If a ... $(3)\;\;\;\text{Niether}\; \mathrm{S_1}\;\text{nor}\;\mathrm{S_2}$

asked Dec 29, 2019 in Set Theory & Algebra Sanjay Sharma 460 views

0 votes

0 answers

7

ISI2015-MMA-92
Consider the group $G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$ ... is of finite order $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).

Consider the group $G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$ ... $G/N$ is of finite order $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).

asked Sep 23, 2019 in Set Theory & Algebra Arjun 151 views

1 vote

0 answers

8

ISI2015-MMA-93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$

Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$

asked Sep 23, 2019 in Set Theory & Algebra Arjun 271 views

0 votes

1 answer

9

ISI2015-MMA-94
Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is $4$ $1$ $2$ $3$

Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is $4$ $1$ $2$ $3$

asked Sep 23, 2019 in Set Theory & Algebra Arjun 150 views

0 votes

1 answer

10

ISI2018-MMA-15
Let $G$ be a finite group of even order. Then which of the following statements is correct? The number of elements of order $2$ in $G$ is even The number of elements of order $2$ in $G$ is odd $G$ has no subgroup of order $2$ None of the above.

Let $G$ be a finite group of even order. Then which of the following statements is correct? The number of elements of order $2$ in $G$ is even The number of elements of order $2$ in $G$ is odd $G$ has no subgroup of order $2$ None of the above.

asked May 11, 2019 in Set Theory & Algebra akash.dinkar12 815 views

1 vote

1 answer

11

ISI2019-MMA-19
Let $G =\{a_1,a_2, \dots ,a_{12}\}$ be an Abelian group of order $12$ . Then the order of the element $ ( \prod_{i=1}^{12} a_i)$ is $1$ $2$ $6$ $12$

Let $G =\{a_1,a_2, \dots ,a_{12}\}$ be an Abelian group of order $12$ . Then the order of the element $ ( \prod_{i=1}^{12} a_i)$ is $1$ $2$ $6$ $12$

asked May 7, 2019 in Set Theory & Algebra Sayan Bose 916 views

0 votes

1 answer

12

ISI-MMA-2015-92
Consider the group $G \;=\; \begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix}\;: a,b \in \mathbb{R},a>0 \end{Bmatrix}$ ... order (D) $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^{+}$(the group of positive reals with multiplication).

Consider the group $G \;=\; \begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix}\;: a,b \in \mathbb{R},a>0 \end{Bmatrix}$ ... is of finite order (D) $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^{+}$(the group of positive reals with multiplication).

asked Mar 6, 2019 in Set Theory & Algebra ankitgupta.1729 387 views

21 votes

5 answers

13

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 $G$: $R_1: \forall a , b \in G, \: a R_1 b \text{ if and only if } \exists g \in G \text{ such that } a = g^{-1}bg$ $R_2: \forall a , b \in G, \: a R_2 b \text{ if and only if } a= b^{-1}$ Which of the above is/are equivalence relation/relations? $R_1$ and $R_2$ $R_1$ only $R_2$ only Neither $R_1$ nor $R_2$

asked Feb 7, 2019 in Set Theory & Algebra Arjun 7.9k views

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