Recent questions tagged group-theory

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason RAssertion A: The order of the group ( $\mathrm{Z}_{5}, \times$ ) divid...

If ( G ) is a group and ( H ) is a subset of ( G ) such that ( H ) is closed under the group operation and every element in ( H ) has an inverse in ( H ), which statement...

Every group of order 4 isCyclicAbelianNon abelianNone of the Above

The inverse of -1 in the multiplicative group, $\{1,-1, i,-i\}$ is1-1 i-i

$\{1, i,-i,-1\}$ is......SemigroupSubgroupCyclic groupAbelian group

Identity element of the group $R^*$ under Multiplication is120-1

Identity element of the group $R$ under addition is120-1

A subgroup $H$ of a group $G$ is normal if:$a H=H a$ for all $a \in G$.$H$ is commutative.$H$ has the same order as $G$.$H$ is the center of $G$.

In a group $G$, an element a has order 2 if:$a^2=a$.$a^2=e$.$a=e$.none of these.

Which of the following groups is not cyclic?$\mathbb{Z} / 6 \mathbb{Z}$$\mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}$$\mathbb{Z} / 4 \mathbb{Z}$$\mathbb{Z}$

Which of the following is a cyclic group?The set of all integers under addition.The set of all real numbers under addition.The set of all integers under multiplication.Th...

What is the identity element of the group of non-zero real numbers under multiplication?01-110

Which of the following sets with the given operation is a group?The set of all integers with addition.The set of all natural numbers with multiplication.The set of all ra...

Which of the following is an example of a group$(R,-)$$(Z,-)$$(R,+)$$(Q,-)$

Which of the following is an abelian group?The group of all permutations of a set.The group of all $\mathbf{2} \times \mathbf{2}$ invertible matrices.The group of all rot...

If $G$ is a group and $g \in G$, the order of $g$ is defined as:The smallest positive integer $n$ such that $g^n=e$.The largest positive integer $n$ such that $g^n=e$.The...

Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statemen...

If $x$ and $y$ are elements in a group $G$ and if $x^{5}=y^{3}=e,$ where $e$ is the identity of $G,$ then the inverse of $x^{2} y x^{4} y^{2}$ must be$y^{2} x y^{2} x^{4}...

Let $G$ be a group of order $n$. Suppose $G$ has no proper nontrivial subgroups (i.e., the only subgroups of $G$ are $\{e\}$ and $G$ itself).Find the smallest possible va...

Let $G$ be a finite group and consider the following statements, which of the following statement are true?If $G$ has even order, then it contains an element $a \neq e$ s...

Let $G$ be a group and consider the following statements:A. If $G$ is abelian, then $(a \cdot b)^n=a^n \cdot b^n$ for all $a, b \in G$ and all integers $n$.B. If $(a \cdo...