Zassenhaus isomorphism theorem
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This article is about an isomorphism theorem in group theory.
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- Normality satisfies transfer condition: If is normal in and is a subgroup of , is normal in .
- Modular property of groups: If , then and .
- Join lemma for normal subgroup of subgroup with normal subgroup of whole group: If and is normal in , then the subgroup is normal in the subgroup .
- Second isomorphism theorem: If are subgroups of a group such that is normal in , then is isomorphic to .
To prove: is normal in and , and we have an isomorphism:
- (Given data used: ; Fact used: fact (1)): is normal in : This follows from fact (1), setting .
- (Given data used: ): and normalize : Since is normal in , the normalizer of contains , hence it also contains .
- and are subgroups: This follows directly from step (2).
- is normal in : By step (2), is normal in , and by step (1), is normal in . Thus, by fact (3), we obtain that is normal in . (Here, we set ).
- (Facts used: fact (4)): Setting and , we observe that by step (4), is normal in , so applying fact (4) yields:
- (Facts used: fact (2)): Applying fact (2) with yields . (Note that both are equal because of the fact that , being normal in , permutes with . We thus have:
- is normal in , both are subgroups, and we have (by analogous reasoning to steps (1) - (6)):
- The right sides for steps (6) and (7) are equal, hence the left sides are isomorphic, completing the proof.