(1). Which of the following groups is isomorphic to the factor group 2/42 ?

(a). Z . (b). Z4. (c). 4E.

((1). None of the above. (2). Let G be a ﬁnite group with IGl = 15, which of the following statements is

false ? a). If H is a subgroup of G, then ‘H‘ is one of {1, 3,5, 15}. b). If a E G, the order of a. is one of {1,3, 5, 15}. c). G has no element with order 7. (1). G has an element with order 15. 3).Let G be a ﬁnite group, which of the following statements is true ?

a). If d is a divisor of G, then G has a subgroup H with \H\ = d. (

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(b). If d is a divisor of G, then G has an element a. such that a. has order d.

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( c). For a, b E G, the order of ab is equal to the order of ba.

(1). For (1,?) E G, we have ab = be. 4). The maximal possible order of an element in 55 is

a) 6, (b). 15, (c). 720 ((1). None of above 5). If R is a commutative ring with unity 1, a E R, a. 75 U is a unit, that is,

there exists I) E R with ab = 1, which of the following is false? (a) a5 7’: 0. (b) a2 is NOT a U-divisor. (c) a. is NOT a 0-divisor. (d) a — 1 is a 0 divisor.