Group Theory PS03EMTH36 (25-10-2021)

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Question 1. Let

G

be a group with

a^{5} = 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

2021

. Then

G

does not have a subgroup of order

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 3. A cycle of length is called a transposition.

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 4. The number of conjugate classes in

S_{4}

is .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 5. A subgroup of order [1.2cm] is called a

2

-sylow subgroup of a group of order

48

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 6. A

p

-Sylow subgroup of a group is normal if it is .

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 7. Let

G_{1}

and

G_{2}

be two groups. If

G_{1} \times G_{2}

is abelian, then abelian.

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

Question 8. The number of invariants of the group

ℤ_{8} \times ℤ_{4}

is .

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

★	right first attempt
✓	right
✗	wrong

	i)	$G$ is cyclic
	ii)	$o (G)$ is even
	iii)	$o (G)$ is odd
	iv)	none of these

	i)	$G_{1}$ or $G_{2}$ is
	ii)	only $G_{2}$ is
	iii)	only $G_{1}$ is
	iv)	$G_{1}$ and $G_{2}$ are

	i)	$1$
	ii)	$2$
	iii)	$3$
	iv)	$4$

	i)	$4$
	ii)	$5$
	iii)	$12$
	iv)	$24$

	i)	$3$
	ii)	$4$
	iii)	$16$
	iv)	$32$

	i)	abelian
	ii)	non-abelian
	iii)	unique
	iv)	simple