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Recent questions tagged group-theory
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61
GO Classes Weekly Quiz 12 | Discrete Mathematics | Group Theory, Functions | Question: 5
Consider the group $0,1,2,3,4,+{ }_{5}$. What will be the value of $2^{-3}$ and $3^{-2}$ for the given group? $1$ and $1$ $2$ and $3$ $4$ and $4$ $3$ and $2$
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GO Classes Weekly Quiz 12 | Discrete Mathematics | Group Theory, Functions | Question: 9
The group $Z_{n}$ consists of the elements $\{0,1,2, \ldots, n-1\}$ with addition $\bmod n$ as the operation. How many subgroups of $Z_{9}$ are there?
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GO Classes Weekly Quiz 12 | Discrete Mathematics | Group Theory, Functions | Question: 10
Let $Z$ be the set of all integers. Let $n \in Z$ and $nZ = \{nk : k \in Z \}.$ We know that $Z$ is a group under addition operation. Which of the following is/are true? $nZ$ is a subgroup of $Z$ (under ... for some $n.$ $3Z$ is the smallest subgroup of $(Z,+)$ containing $3.$ $nZ$ is a cyclic subgroup of $Z.$
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GO Classes Weekly Quiz 12 | Discrete Mathematics | Group Theory, Functions | Question: 11
The set $Z_{\mathrm{n}}^{*}$ consists of the elements $\{1,2, \ldots, \mathrm{n}-1\}$ with multiplication $\bmod n$ as the operation. The group $Z_{n}$ consists of the elements $\{0,1,2, \ldots, n-1\}$ ... then $\mathrm{G}$ is a cyclic group. The number of elements in $Z_{8}^{*}$ which have an inverse is $4.$
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GO Classes Weekly Quiz 12 | Discrete Mathematics | Group Theory, Functions | Question: 12
Consider the standard groups $Z_{\mathrm{n}}, \mathrm{R}^{*}, \mathrm{C}^{*}$ under their standard group operation. Here, $\mathrm{R}^{*}, \mathrm{C}^{*}$ are set of non-zero real numbers, set of non-zero complex numbers, ... $\sqrt{3} \in \mathrm{R}^{*}$ $-\mathrm{i} \in \mathrm{C}^{*}$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 2
Let $\text{G}$ be a cyclic group with generator ‘$a$’. Order of ‘$a$’ is $29.$ The number of subgroups $\text{G}$ has __________
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 3
$(\text{Q}, \ast)$ is an algebraic structure where $\text{Q}$ represents rational numbers and $\ast$ denotes multiplication. Which one of the following statements is true? $\text{Q}$ is an abelian group. $\text{Q}$ is a group but not abelian. $\text{Q}$ is a semigroup but not a monoid. $\text{Q}$ is monoid but not group.
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 5
Given an operation table for group $\text{G}(\{1, 2, 3, 4, 5\})$ then what will be the identity element ‘$e$’ and inverse of ‘$3$’. $e = 1\;\&\; 3^{-1} = 4$ $e = 5\;\&\; 3^{-1} = 1$ $e = 1\;\&\; 3^{-1} =2$ $e = 1\;\&\; 3^{-1} = 5$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 6
If there is a group $\text{G}(z, +)$ where $z$ refers to the set of integers, and “$+$” is addition. Then which of the following are possible sub groups. (set of even numbers $, +)$ (multiples of $3, +)$ (set of odd numbers $, +)$ (multiples of $5, +)$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 8
If the order of a group is $36$ then which of the following is a possible order of an element of a group. $5$ $7$ $6$ $10$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 9
Let there be a semigroup $(\text{G}, \ast)$ where $\text{G} = {p, q, r,s}$ such that $p \ast q = r ; q \ast q = p$ and $s \ast p = q.$ Then $s \ast r = ?$ $p$ $q$ $r$ $s$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 10
Which of the following is/are NOT a group? The integers under addition The nonzero integers under multiplication The nonzero real numbers under multiplication The complex numbers under addition
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 11
What is the order of the element $a^{32}$ in the cyclic group $G = {1, a, a^{2} , \dots, a^{37}}.$ Note that $\text{G}$ is a cyclic group of order $38$ with generator $a. 1$ is the identity element.
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 12
Let $\text{G}$ be a group. Let $g,h,x,y \in \text{G}$ be given. It is given that $x \ast g = h, y \ast g = h$ $x = y$ $x \ast x \ast x = y \ast y \ast y$ $x \ast y \ast x = y \ast x \ast y$ Only I Only II Only I and III All of I, II and III
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 15
Let $\text{G}$ be a group under binary operation $\ast .$ Let $g \in \text{G}.$ We define $\langle g \rangle$ as follows : $\langle g \rangle = \{g^{n} \mid n \in \mathbb{Z}\}.$ ... abelian. If $\text{H} \leq \text{G}$ and $g \in \text{H},$ then $ \langle g \rangle \leq \text{H}.$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 16
Consider the group $(\mathbf{Z}, +)$ where $\mathbf{Z}$ is set of all integers and $+$ is addition operation. Consider the following subset of $\mathbf{Z}:$ $\text{H} = {30x + 42y + 70z | x, y, z \in \mathbf{Z}}.$ ... of $\mathbf{Z}$ because for some element $g$ in $\text{H},$ inverse of $g$ doesn't belong to $\text{H}.$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 19
Which of the following are abelian group $\forall x, y \in$ integer? $x \ast y = x + y$ $x \ast y = \max (x, y)$ $x \ast y = x × y$ $x \ast y = x^{2} + y^{2}$
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 20
Let $\text{R}$ be the set of all real numbers. Let $\text{S} = \text{R}\setminus \{-1\}$ and define a binary operation on $S$ by $a \ast b = a + b + ab.$ Which of the following is true? $(\text{R},\ast)$ is an ... $(\text{R},\ast),(\text{S},\ast)$ are abelian groups.
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GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 21
In a group $\text{G},$ every element other than the identity element has order $2$ then $\text{G}$ is? Abelian group Cyclic group Non-abelian group Non-cyclic group
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