Group Theory PS03EMTH36 (03-11-2018)

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Question 1.

ϕ (

) = ϕ (

)

, where

ϕ

denotes Euler’s totient function.

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Incorrect.

Choice iv) is Correct!

Question 2. If

N

is a subgroup of a group

G

such that

G ∕ N

is a group, then

N

is .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 3. Order of automorphism group

𝒜 (ℤ)

ℤ

, is .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 4. If

e \neq a \in

, then

c_{a} \neq 1

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 5. A group of order is not simple.

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 6. The order of

7

-Sylow subgroups of

S_{25}

is .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

Question 7. A

p

-Sylow subgroup

H

of a group

G

is unique if

H

Choice i) is Incorrect.

Choice ii) is Incorrect.

Choice iii) is Correct!

Choice iv) is Incorrect.

Question 8. Number of nonisomorphic abelian groups of order

128

is .

Choice i) is Incorrect.

Choice ii) is Correct!

Choice iii) is Incorrect.

Choice iv) is Incorrect.

★	right first attempt
✓	right
✗	wrong

	i)	cyclic
	ii)	kernel of a homomorphism.
	iii)	commutative
	iv)	none of these can be said

	i)	$2, 6$
	ii)	$5, 9$
	iii)	$7, 11$
	iv)	$8, 12$

	i)	$S_{2}$
	ii)	$S_{7}$
	iii)	$ℤ_{3}$
	iv)	$ℤ$

	i)	is nonabelian
	ii)	is abelian
	iii)	$\subset Z (G)$
	iv)	is finite

	i)	$5$
	ii)	$15$
	iii)	$20$
	iv)	$64$