If G is a group and H is a subgroup of index 2 in G then choose the correct statement.a)H is a normal subgroup of Gb)H is not a normal subgroup of Gc)H is a subgroup of Gd)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

To understand why option A is the correct answer, we need to first understand the concept of a normal subgroup.

A subgroup H of a group G is said to be a normal subgroup if and only if for every element g in G, the conjugate of H by g is also a subset of H. In other words, for any h in H and g in G, the element ghg^(-1) is also in H.

Now, let's consider the given information: G is a group and H is a subgroup of index 2 in G.

What does it mean for H to have index 2 in G?
The index of a subgroup H in G is the number of distinct left cosets of H in G. In this case, since H has index 2 in G, there are exactly two distinct left cosets of H in G.

Proof that H is a normal subgroup of G:
Since H has index 2 in G, there are two distinct left cosets of H in G, denoted by H and gH, where g is an element of G but not in H.

Consider an element h in H and an arbitrary element g in G. We want to show that the element ghg^(-1) is also in H.

Case 1: gh is in H
If gh is in H, then ghg^(-1) is also in H since H is a subgroup. In this case, H is a normal subgroup of G.

Case 2: gh is not in H
If gh is not in H, then gh is in the coset gH. Since there are only two distinct cosets, the other coset must be H itself. Therefore, gH = H, which implies that g is in H.

Now, consider ghg^(-1). Since g is in H, we can write g = h' for some h' in H. Therefore, ghg^(-1) = h'hg^(-1). Since H is a subgroup, h'h is also in H. Thus, ghg^(-1) is in H.

In both cases, we have shown that for any h in H and any g in G, ghg^(-1) is in H. Therefore, H is a normal subgroup of G.

Hence, the correct statement is option A: H is a normal subgroup of G.

Answer

When order of subgroup divides order of group, and if their index is 2 then H is normal in G

Answer

H is a subgroup then if order group divided by order subgroup gives you 2 then it is normal subgroup
Formula is o(G) /o(H) =index 2 =>normal subgroup

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