If G is a group of order 23 then find total no. of subgroups of group G.Correct answer is '2'. Can you explain this answer?

Question

Answer

Since 23 is prime, it's divisors are 1 and 23. Hence if H ⊆ G then by lagrange's theorem O(H) divides O(G).

So that 0(H) = 1 or 23.

if O(H) = 1 => H ={e}

if O(H) = 23 =>H = G

Which shows that G has only trivial subgroups.

Total no. of subgroups are 2.

Answer

Explanation:

Order of a group:
- The order of a group is the number of elements in the group.
- In this case, the order of group G is 23.

Subgroups of a group:
- A subgroup of a group is a subset of the group that is itself a group under the same operation.
- There are two trivial subgroups in any group: the group itself and the trivial subgroup {e} containing only the identity element.

Proof:
- By Lagrange's theorem, the order of a subgroup must divide the order of the group.
- Therefore, the possible orders of subgroups of a group of order 23 are 1 and 23.
- There is only one subgroup of order 23, which is the group itself.
- There is also one subgroup of order 1, which is the trivial subgroup {e}.

Total number of subgroups:
- Therefore, the total number of subgroups of a group of order 23 is 2: the group itself and the trivial subgroup.

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