Maximal implies normal or abnormal
This article gives the statement, and possibly proof, of a result according to which any Maximal subgroup (?) of a group satisfies exactly one of the following two subgroup properties: Normal subgroup (?) and Abnormal subgroup (?)
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Further information: Maximal subgroup
Further information: Normal subgroup
A subgroup of a group is termed normal in if it satisfies the following equivalent conditions:
- For any , the subgroup is equal to .
- The normalizer of in equals .
Further information: Abnormal subgroup
A subgroup of a group is termed abnormal in if, for any , where .
Given: A group , a maximal subgroup of .
To prove: is either normal or abnormal in .
Proof: Let be the normalizer of in . Then, . Thus, either , or . In the former case, is normal in .
In the latter case, . Now, pick any . Consider the subgroup . There are three cases:
- is not contained in : By maximality of , , so .
- : Thus, , and since , we get . Thus, .
- is a proper subgroup of : is a proper subgroup of , forcing , which would imply that , a contradiction.