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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Consider the following statements:
A. A group can have zero elements.
B. A group cannot have any element which is idempotent except the identity element.
C. An isomorphism from a group $\langle G,\rangle$ to $\langle G,\rangle$ is called an automorphism.
D. If every element in a group is its own inverse, then the group must be abelian.
Choose the correct answer about the four statements given above.
Solution
Statement A is false.
A group cannot have zero elements because every group must contain an identity element.
Statement B is true.
If $a$ is idempotent, then
$a*a=a$
Multiplying by $a^{-1}$,
$a=e$
So, only identity element is idempotent.
Statement C is true.
An isomorphism from a group to itself is called an automorphism.
Statement D is true.
If every element is its own inverse, then
$a^{-1}=a$ and $b^{-1}=b$
Now,
$(ab)^{-1}=b^{-1}a^{-1}=ba$
But every element is its own inverse, so
$(ab)^{-1}=ab$
Therefore,
$ab=ba$
Hence, the group is abelian.
So, correct option is:
$\boxed{(a)\ \text{B, C and D are correct}}$
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