Question

Which of the following is a subgroup of $a=\{0,1,2,3,4,5\}$, under addition modulos $6?$

• A. $\{0,2\}$
• B. $\{0,1\}$
• C. $\{0,4\}$
• D. $\{0,3\}$

Answer 1

The correct option is D $\{0,3\}$

If any group is a subgroup of a group then it should follow the conditions of the parent group.

Given parent group: $G\equiv \{0,1,2,3,4,5\}$

We have to find a subgroup which follows the conditions under addition modulo $6$.

Lets make Caylay table for each option.

A) $\{0,2\}$

	$0$	$2$
$0$	$0$	$2$
$2$	$2$	$4$

In this the subgroup does not contains the element $4$.

Therefore, it is not the subgroup of $G$

B) $\{0,1\}$

	$0$	$1$
$0$	$0$	$1$
$1$	$1$	$2$

In this the subgroup does not contains the element $2$.

Therefore, it is not the subgroup of $G$

C) $\{0,4\}$

	$0$	$4$
$0$	$0$	$4$
$4$	$4$	$2$

In this the subgroup does not contains the element $2$.

Therefore, it is not the subgroup of $G$

D) $\{0,3\}$

	$0$	$3$
$0$	$0$	$3$
$3$	$3$	$0$

In this, the subgroup does contains the element $0$ and $3$.

Therefore, it is the subgroup of $G$