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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Consider the following statements:
$S_1:$ If a group $(G,*)$ is of order $n$, and $a\in G$ is such that $a^m=e$ for some integer $m\le n$, then $m$ must divide $n$.
$S_2:$ If a group $(G,)$ is of even order, then there must be an element $a\in G$ such that $a\ne e$ and $aa=e$.
Which of the statements is/are correct?
Solution
For $S_1$:
By Lagrange theorem, the order of every element of a finite group divides the order of the group.
So, if $m$ is the order of element $a$, then $m$ divides $n$.
Hence, $S_1$ is true.
For $S_2$:
If a finite group has even order, then there exists at least one non-identity element of order $2$.
So, there exists $a\ne e$ such that:
$a*a=e$
Hence, $S_2$ is also true.
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