group-theory

Let Zₙ be the group of integers {0, 1, 2, ..., n-1} with addition modulo n as the group operation. The number of elements in the group Z₂ × Z₃ × Z₄ that are their own inverses is _...

Let Z n be the group of integers {0, 1, 2, ..., n − 1} with addition modulo n as the group operation. The number of elements in the group Z 2 × Z 3 × Z 4 that are their own inverse...

Let X be a set and 2$$^X$$ denote the powerset of X. Define a binary operation $$\Delta$$ on 2$$^X$$ as follows: $$A\Delta B=(A-B)\cup(B-A)$$. Let $$H=(2^X,\Delta)$$. Which of the...

Which of the following statements is/are TRUE for a group G?

Let G be a group order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?

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 finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.

let $$G$$ be a group with $$15$$ elements. Let $$L$$ be a subgroup of $$G$$. It is known that $$L \ne G$$ and that the size of $$L$$ is at least $$4$$. The size of $$L$$ is ______.

There are two elements $$x, y$$ in a group $$\left( {G,\, * } \right)$$ such that every elements in the group can be written as a product of some number of $$x's$$ and $$y's$$ in s...

Consider the set $$S = \left\{ {1,\,\omega ,\,{\omega ^2}} \right\},$$ where $$\omega $$ and $${{\omega ^2}}$$, are cube roots of unity. If $$ * $$ denotes the multiplication opera...

Which one of the following in NOT necessarily a property of Group?

How many different non-isomorphic Abelian groups of order 4 are there?

The set $$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$ under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is fal...

The set $$\left\{ {1,\,\,2,\,\,4,\,\,7,\,\,8,\,\,11,\,\,13,\,\,14} \right\}$$ is a group under multiplication modulo $$15$$. The inverse of $$4$$ and $$7$$ are respectively:

Consider the set $$H$$ of all $$3$$ $$X$$ $$3$$ matrices of the type $$$\left[ {\matrix{ a & f & e \cr 0 & b & d \cr 0 & 0 & c \cr } } \right]$$$ Where $$a, b, c, d, e$$ and $$f$$...

Let $$S = \left\{ {0,1,2,3,4,5,6,7} \right\}$$ and $$ \otimes $$ denote multiplication modulo $$8$$, that is, $$x \otimes y = \left( {xy} \right)$$ mod $$8$$ (a) Prove that $$\left...

Which one of the following is false?

Let A be the set of all nonsingular matrices over real numbers and let * be the matrix multiplication operator. Then

Let $${G_1}$$ and $${G_2}$$ be subgroups of a group $$G$$. (a) Show that $${G_1}\, \cap \,{G_2}$$ is also a subgroup of $$G$$. (b) $${\rm I}$$s $${G_1}\, \cup \,{G_2}$$ always a su...

Some group (G, o) is known to be abelian. Then, which one of the following is true for G?

(a) If G is a group of even order, then show that there exists an element $$a \ne e$$, the identifier $$g$$, such that $${a^2} = e$$ (b) Consider the set of integers $$\left\{ {1,2...