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Proving that a subgroup is normal - Groupprops

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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. In some cases, proving that a certain subgroup is normal may be hard or impossible, perhaps because the subgroup is not normal. The following alternative approaches are useful here: The best way to try proving that a subgroup is normal is to show that it satisfies one of the standard equivalent definitions of normality. To prove that is a normal subgroup of , we can construct a homomorphism such that the kernel of the homomorphism, i.e., the set of elements that map to the identity, is precisely . Note that if we do this successfully, it is not even necessary to establish separately that is a subgroup. Here are some examples: Another way of proving normality is using the inner automorphism, or conjugation, definiti

Proving that a subgroup is normal - Groupprops Jump to content Want site search autocompletion? See here Encountering 429 Too Many Requests errors when browsing the site? See here From Groupprops This survey article is about proof techniques for or related to satisfaction of the following property: normal subgroup Find other survey articles about normal subgroup | Find fact articles that prove satisfaction of this property 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 i

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