# Recent questions tagged group-theory

1
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
2
Which one of the following is NOT necessarily a property of a Group? Commutativity Associativity Existence of inverse for every element Existence of identity
1 vote
3
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
4
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
5
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 ... $(3)\;\;\;\text{Niether}\; \mathrm{S_1}\;\text{nor}\;\mathrm{S_2}$
6
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).
7
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? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$
8
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 $4$ $1$ $2$ $3$
9
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.
1 vote
10
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 $1$ $2$ $6$ $12$
11
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).
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 $R_2$