If G =({0,1,2,3,4,5,6,7} ,⊕₈) is a group. Which of the following are correct for subgroups of G?
a) {2,5}
b) {0,2}
c) {0,4}
d) {0,2,4}
e) {0,2,4,6}

(A) Only b and c
(B) Only b, c, and d
(C) only c and e
(D) Only a, b, c, and e


Answer: (C)

Explanation: Check one by one all subgroups.
a) Lagrange’s Theorem -> states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G.
8/2 = 4. Not violating Lagrange’s Theorem Check further for Closure.

It is not closed on ⊕₈. So it cannot be the subgroup of G.

b) 8/2=4. Not violating Lagrange’s Theorem Check further for Closure.

It is not closed under ⊕₈. So it cannot be the subgroup of G.

c) 8/2=4. Not violating Lagrange’s Theorem Check further for Closure.

It is closed under ⊕₈. So it subgroup of G.

d) 8/3= not divisible. Violating Lagrange’s Theorem. So it can not be the subgroup of G.

We will find closure also for this.
It is not closed under ⊕₈. So it cannot be the subgroup of G.

e) 8/4=2. Not violating Lagrange’s Theorem Check further for Closure

It is closed under modulo ⊕₈. So it is the subgroup of G.

So Only c and e.


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