Group Theory PS03EMTH36 (01-01-2021)

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Question 1. Let

G

be a group with

a^{- 1} = a

for all

a \in G

. Then .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 2. Let

G

be a group of order

72

. Let

H

and

K

be subgroups of

G

of order

9

and

4

respectively. Then

o (H K) =

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 3. Let

G

be a group of order

p^{2}

. Which of the following need not be true?

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 4. Which of the following is the class equation of a group of order

4

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

Question 5. The order of

2

-sylow subgroup of

S_{8}

is .

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

Question 6. A group of order is simple.

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 7. Let

A

and

B

be two groups. Then

A \times B

is abelian if

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 8. The invariants of a non-cyclic group of order

121

is/are .

Choice i) is Correct!

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Incorrect.

★	right first attempt
✓	right
✗	wrong

	i)	$G$ is cyclic
	ii)	$G$ is abelian
	iii)	$o (G) = 2$
	iv)	none of these

	i)	$G$ has an element of order $p$
	ii)	$G$ has a subgroup of order $p$
	iii)	$G$ has an element of order $p^{2}$
	iv)	none of these

	i)	only $A$ is abelian
	ii)	only $B$ is abelian
	iii)	both $A$ and $B$ are abelian
	iv)	none of $A$ and $B$ are abelian

	i)	$1$
	ii)	$2$
	iii)	$36$
	iv)	$18$

	i)	$1, 1$
	ii)	$11, 11$
	iii)	$2, 2$
	iv)	$2$