(D) subset $H$ of a group $G$ is a subgroup if it is itself a group under the same operation.For $H = \{4^{n} \mid n \in \mathbb{Z}\}$,we note that $4

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

$\{4^{n} \mid n \in \mathbb{Z}\}$

(D) subset $H$ of a group $G$ is a subgroup if it is itself a group under the same operation.For $H = \{4^{n} \mid n \in \mathbb{Z}\}$,we note that $4^{n} = (2^{2})^{n} = 2^{2n}$. Since $2n \in \mathbb{Z}$ for all $n \in \mathbb{Z}$,$H \subset G$.$1.$ Closure: $4^{n} \cdot 4^{m} = 4^{n+m} \in H$.$2.$ Identity: $4^{0} = 1 = 2^{0} \in H$.$3.$ Inverse: For $4^{n} \in H$,the inverse is $4^{-n} \in H$.Thus,$\{4^{n} \mid n \in \mathbb{Z}\}$ is a subgroup of $G$.

Class 12 Mathematics Relations and Functions Binary Operation

Medium

Which of the following is a subgroup of the group $G = \{2^{n} \mid n \in \mathbb{Z}\}$ under multiplication?

KCET 2012 All KCET

A
$\{4^{n} \mid n \in \mathbb{N}\}$
B
$\{3^{n} \mid n \in \mathbb{Z}\}$
C
$\{6^{n} \mid n \in \mathbb{N}\}$
D
$\{4^{n} \mid n \in \mathbb{Z}\}$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

Relations and Functions

Subtopic

Binary Operation

Previous Year

KCET 2012 Paper All KCET years →

Determine whether or not each of the definitions of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^+$,define $$ by $a b = |a - b|$.

Medium

View Solution

Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

Medium

View Solution

We define a binary relation $\sim$ on the set of all $3 \times 3$ real matrices as $A \sim B$ if and only if there exist invertible matrices $P$ and $Q$ such that $B = P A Q^{-1}$. The binary relation $\sim$ is

MediumWBJEE 2014

View Solution

Determine whether or not the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a - b$.

Medium

View Solution

Let $$ be a binary operation defined on $R$ by $a b = \frac{a+b}{4}$ for all $a, b \in R$. Then the operation $*$ is:

DifficultKCET 2016

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

Which of the following is a subgroup of the group $G = \{2^{n} \mid n \in \mathbb{Z}\}$ under multiplication?

Similar Questions

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^+$,define $*$ by $a * b = |a - b|$.

Show that $*: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a * b = a + 2b$ is not commutative.

We define a binary relation $\sim$ on the set of all $3 \times 3$ real matrices as $A \sim B$ if and only if there exist invertible matrices $P$ and $Q$ such that $B = P A Q^{-1}$. The binary relation $\sim$ is

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a - b$.

Let $*$ be a binary operation defined on $R$ by $a * b = \frac{a+b}{4}$ for all $a, b \in R$. Then the operation $*$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Determine whether or not each of the definitions of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^+$,define $$ by $a b = |a - b|$.

Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

Determine whether or not the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a - b$.

Let $$ be a binary operation defined on $R$ by $a b = \frac{a+b}{4}$ for all $a, b \in R$. Then the operation $*$ is: