search
Log In
0 votes
699 views

Let $G$ be a finite group of even order. Then which of the following statements is correct?

  1. The number of elements of order $2$ in $G$ is even
  2. The number of elements of order $2$ in $G$ is odd
  3. $G$ has no subgroup of order $2$
  4.  None of the above.
in Set Theory & Algebra 699 views

1 Answer

1 vote
Answer is B.

Since the group is of even order and the identity is the inverse of itself, therefore, there are odd number of elements (other than identity).

Also, if there is some element which is not inverse of itself, then we can pair such elements with their inverses. This will leave us with odd number of elements which have to be their own inverses.
0

here why did you not include identity

0
unable to understand example?

Related questions

3 votes
1 answer
1
584 views
Let $C_i(i=0,1,2...n)$ be the coefficient of $x^i$ in $(1+x)^n$.Then $\frac{C_0}{2} – \frac{C_1}{3}+\frac{C_2}{4}-\dots +(-1)^n \frac{C_n}{n+2}$ is equal to $\frac{1}{n+1}\\$ $\frac{1}{n+2}\\$ $\frac{1}{n(n+1)}\\$ $\frac{1}{(n+1)(n+2)}$
asked May 11, 2019 in Combinatory akash.dinkar12 584 views
1 vote
1 answer
2
484 views
A new flag of ISI club is to be designed with $5$ vertical strips using some or all of the four colors: green, maroon, red and yellow. In how many ways this can be done so that no two adjacent strips have the same color? $120$ $324$ $424$ $576$
asked May 11, 2019 in Combinatory akash.dinkar12 484 views
1 vote
1 answer
3
869 views
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 $1$ $2$ $6$ $12$
asked May 7, 2019 in Set Theory & Algebra Sayan Bose 869 views
2 votes
1 answer
4
1.4k views
Let $H$ be a subgroup of group $G$ and let $N$ be a normal subgroup of $G$. Choose the correct statement : $H\cap N$ is a normal subgroup of both $H$ and $N$ $H\cap N$ is a normal subgroup of $H$ but not necessarily of $N$ $H\cap N$ is a normal subgroup of $N$ but not necessarily of $H$ $H\cap N$ need not to be a normal subgroup of either $H$ or $N$
asked Apr 24, 2018 in Set Theory & Algebra Tesla! 1.4k views
...