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Definition: Subgroup

A subgroup is a substructure $H\subseteq G$ of a group $(G,\ast)$, which is itself a group. In particular, $H$ fulfills the definition of a group:

1. $e\in H\cap G$, where $e$ is the unique neutral element of $G.$
2. If \(a\in H\cap G\) then \(a^{-1}\in H\), where $a^{-1}$ denotes each the unique inverse element of $a.$
3. If \(a\in H\cap G\) then \(a^{-1}\in H\), where $a^{-1}$ denotes each the unique inverse element of $a.$

Mentioned in:

Definitions: 1 2 3 4 5 6
Examples: 7
Explanations: 8
Lemmas: 9 10 11 12 13
Problems: 14
Proofs: 15 16 17 18 19 20 21 22 23 24 25 26
Propositions: 27 28 29 30
Solutions: 31
Theorems: 32

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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001