group theory

Question

which among the following statements is TRUE ?

S1 : ( { 0,1,2....(m-1) } , +m ) where +m stands for "addition-modulo-m"

S2 : ( {0,1,2....m} , +m ) where +m stands for "addition-modulo-m".

A) ONLY S1 is a group.

B) ONLY S2 is a group.

C) BOTH S1 AND S2 are groups.

D) NEITHER S1 NOT S2 is a group.

Answer 1

The answer will be A. Only $S1$ is the group.

Because in the second case, Identity element does not exist.

In the first case identity element is $0$. That means for all $ a \in S1 $, $ a +_{m} 0 = a $. because $a< m$

In the second case, $0$ can not be the identity element. For example: for one of the member $m$ of the set we have $ m +_{m} 0 = 0 $ , It should come $m$. That's why $S2$ is not a group.

However, both $S1$ and $S2$ are Semigroup as they satisfy closure and associativity property.