Group

If $G$ is a group of even order then

In a group $\left(G,*\right)$ if there exists an element $b$ such that $b^{*}a=b$ for all $a\in G\mathit{,}$ then order of the group is

If $\left(G{,}^{*}\right)$ is a group such that $(a* b{)}^2={\left(a^{*}a\right)}^{*}\left(b^{*} b\right)$ for all $a, b, \in G,$ then $G$ is

In the group of non-zero rational numbers under the binary operation $*$ given by  $a{}^{*} b=\frac{ab}{5}$ the identity element and the inverse of $8$ are respectively.

The set of all non-zero real number with the operation defined on it by a ${}^{*} b=\frac{ab}{2}$ is an abelian group. The identity of the group is

If $\left(G{,}^{*}\right)$ is a group such that $a^{*} b=b^{*}a$ for two elements $a$ and $b,$ then

In a group $(G,*)$ if ${\left(a^{*}b\right)}^{-1}=a^{-1}*b^{-1}$ for all $a, b\in G,$ then $G$ is

In a group $\left(G{,}^{*}\right)$ the equation $x*a=b$ has a

$G$ is a set of all rational numbers except $-1$ and by a ${}^{*}b=a+b+ab$ for all $a,b\in G·$ In the group $(G,*)$ the solution of $2^{-1}*x*3^{-1}=5$ is

Let $G$ denote the set of all $n\times n$ non-singular matrices with rational numbers as entries. Then under multiplication

If every element of a group $G$ is its own inverse, then $G$ is

$Z$ is the set of integers, $\left(Z{,}^{*}\right)$ is a group with $a^{*}b=a+b+1, a, b,\in G$ . Then, inverse of $a$ is