- #1
mathmajor2013
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Let G be a group and H be a subgroup of G. Prove that if [G]=2, then H is a normal subgroup of G.
In group theory, [G:H] represents the index of a subgroup H in a group G. It is the number of cosets of H in G. When [G:H]=2, it means that there are only two distinct cosets of H in G.
A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group. This means that for any element g in the group G and any element h in the subgroup H, the element ghg-1 is also in H. In other words, the left and right cosets of a normal subgroup are equal.
Since [G:H]=2, there are only two distinct cosets of H in G. This means that the left and right cosets are equal, and therefore, H is invariant under conjugation by elements of G. Therefore, H is a normal subgroup.
Yes, if [G:H]=2, then H is guaranteed to be a normal subgroup. However, it is not a necessary condition. There are cases where a subgroup may be normal even if [G:H] is not equal to 2.
Yes, consider the group G = {1, 2, 3, 4, 5, 6} under addition modulo 6. Let H = {1, 3}. The index of H in G is [G:H]=2, but H is not normal in G since 2H = {2, 4} and 2H ≠ H2 = {3, 5}.