23.

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29.

In the finite group G={2,4,6,8} with respect to multiplication modulo 10, the identity
element is :

(A) 2 ‏رم‎ 4

(C) 6 0) 8

Which of the following is true?
@ Every group of order 42 has a normal subgroup of order 7
(ii) Every group of order 42 has a normal subgroup of order 8

(A) _ both (i) and (ii) are true (B) _ neither (i) nor (ii) are true

(0 only (i) is true (D) _ only (ii) is true
Every group of even permutations in S,, can be written as a product of ——————— number
of transpositions:

(A) 1 ‏رم‎ 2n+3

(0) 2n+4 ൯) 2n+8

Consider the statements :

@ Number of homomorphisms from Zio to Zy5 is 5

Gi) Number of homomorphisms from Zo to Zo2 is 4

Gii) Number of homomorphisms from Zi¢ to Zig is 2

(Gv) Number of non-trivial homomorphisms from 713 10 19 18 1

(A) all the statements are true ൯) only @) and (ii) are true

(0) only (i), (i) and (iii) are true (7) only 0), (1) and (iv) are true
The largest possible order of any element in Aut (200) is :

(A) 200 (B) 100

(C) 20 (D) 1

Consider the statements :
@ No group of order 96 is simple
Gi) No group of order 160 is simple

Then
(^) Neither (i) nor (ii) is true (B) Both (i) and (ii) are true
(0 @) is true and (ii) is false (D) (i) is false and (ii) is true

Which of the following is true?
0) ~The set ട്‌; ۰ be 3 is a subring of the ring of M, of 2 x 2 matrices over

integers
Gi) Anelement xe R is called idempotent if x? =x
(ii) Let R be a ring such that x° =x for all xe R, then R is commutative

(iv) Let R be a ring of 2 x 2 matrices over reals, then S = 6 3 :x isa real numbers not
x x
a subring of R

(^) All the statements are true ൯) Only 6), (1) and (iii) are true
(C) Only (0) and (ii) are true (D) Only Gi) and (iii) are true

7 083/2023
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