Maximal implies modular
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., maximal subgroup) must also satisfy the second subgroup property (i.e., modular subgroup)
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Statement with symbols
Suppose is a maximal subgroup of . Then, if are subgroups of such that , we have:
Given: A group , a maximal subgroup .
To prove: If are subgroups of such that , then:
Proof: Since is maximal, either or . We consider both cases.
If , then , so the left side is . contains , so the right side is . Thus, the identity holds for .
If , then both sides are , and again the identity holds.