Group Theory PS03EMTH36 (01-04-2019)

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Question 1. The group is abelian.

Choice i) is Correct!

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 2. Let

G

be a finite group and

a \in G

. Then

a^{2 o (G)} =

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 3. The number of odd permutations in

S_{5}

is .

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 4. is a valid class equation of a group of order

5

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

Question 5. If

G

is a group and

p

is a prime such that

p ∣ o (G)

, then the number of

p

-Sylow subgroups of

G

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

Question 6. The permutation group

S_{4}

cannot have a -Sylow subgroup.

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 7. The number of non-isomorphic abelian groups of order

108

is .

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 8. The invariants of Klein-

4

group are .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

★	right first attempt
✓	right
✗	wrong

	i)	$5 = 1 + 4$
	ii)	$5 = 2 + 3$
	iii)	$5 = 1 + 2 + 2$
	iv)	$5 = 1 + 1 + 1 + 1 + 1$

	i)	is always more than $p$
	ii)	is always less than $p$
	iii)	divides $p$
	iv)	does not divide $p$

	i)	$S_{2}$
	ii)	$S_{3}$
	iii)	$S_{5}$
	iv)	$S_{6}$

	i)	$a$
	ii)	$a^{2}$
	iii)	$e$
	iv)	none of these

	i)	$5$
	ii)	$30$
	iii)	$60$
	iv)	$120$

	i)	$2$
	ii)	$3$
	iii)	$5$
	iv)	$8$