The discussion revolves around proving that a subgroup H of a group G is normal, given that the index |G/H| equals 2. Participants are exploring the implications of this condition on the structure of the group and its cosets.
Some participants have provided insights into the relationship between cosets and the normality of the subgroup. There is an ongoing inquiry into the foundational reasoning behind the established relationships, particularly concerning the identity element and its role in the proof.
Participants are navigating the definitions and properties of groups and subgroups, particularly focusing on the implications of the index of a subgroup. There is a noted interest in understanding the underlying principles rather than just the procedural aspects of the proof.
mathusers said:[G/H]=2 means that H has two left and right cosets in G. Assume g_1 \in H, then it is trivial that g_1H = H = Hg_1. Now assume g_2 \notin H, this means that g_2H \neq H \neq Hg_2. But since there are only 2 cosets and both of them are not in H then it means they are the same so g_2H = Hg_2
is this correct?