Proving that a subgroup is normal
This is a survey article related to:normality
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This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can try proving that the subgroup is normal (or that it is not normal). We first discuss the leading general ideas, and then plunge into the specific cases.
The deviation method of proving normality
In measuring deviation from normality, we see three ways of measuring the extent to which a subgroup deviates from normality: the normalizer, the normal closure and the normal core. Here, we explore each of these as a tool for trying to prove normality.
Normal core
The idea behind using the normal core to prove normality is to show that the given subgroup equals its normal core. In other words, we try to establish that the intersection of all conjugates of the subgroup equals the subgroup itself
Normal closure
The idea behind using the normal closure in order to prove normality is to prove that the subgroup equals its own normal closure. In other words, we show that the subgroup equals that subgroup generated by all its conjugates.
Normalizer
The idea behind using the normalizer to prove normality is to prove that the normalizer of the subgroup equals the whole group. In other words, we show that every element of the group commutes with the subgroup.
Setting up the group action
Let us explore the normal core method a bit further. This method is particularly effective when we are dealing with subgroups of fairly small index.
Let be a subgroup of . Then, acts on the coset space of . This gives a homomorphism from to the symmetric group on the coset space, and the kernel of the homomorphism is the normal core. Hence, the quotient group sits as a subgroup of the symmetric group .
We may sometimes be able to use this to show that (using order considerations. For instance, this is the idea behind showing that a subgroup of least prime index is normal.
