GATE 2006 question on Group

The set {1,2,3,5,7,8,9} under multiplication modulo 10 is not a group. Given below are four possible reasons. Which one of them is false?

(A) It is not closed

(B) 2 does not have an inverse
(C) 3 does not have an inverse
(D) 8 does not have an inverse
 

Answer

Things you need to know

Refer: http://www.techtud.com/short-notes/algebraic-structure

 

Now, 

Let S = {1,2,3,5,7,8,9}

(A) is True. Because, (8*7)%10 = 6 ∉ S

(B) is True. Because, inverse 'e' should exit such that (2*e) % 10 = 1, but such 'e' is not in S.

(C) is False. Because, (3*7)% 10 = 1, Hence, inverse of 3 = 7

(D) is True. Because, inverse 'e' should exit such that (8*e) % 10 = 1, but such 'e' is not in S.

Ans. (C)

 

 

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