User:Ascii/ProofWiki Sampling Notes for Theorems/Group Theory

Index Laws

1. Cancellable Finite Semigroup is Group

   Let $\struct {S, \circ}$ be a non-empty finite semigroup in which all elements are cancellable.

   Then $\struct {S, \circ}$ is a group.

2. Cancellable Infinite Semigroup is not necessarily Group

   Let $\struct {S, \circ}$ be a semigroup whose underlying set is infinite.

   Let $\struct {S, \circ}$ be such that all elements of $S$ are cancellable.

   Then it is not necessarily the case that $\struct {S, \circ}$ is a group.

3. Index Laws/Sum of Indices/Semigroup

   $\forall m, n \in \N_{>0}: a^{n + m} = a^n \circ a^m$

4. Index Laws/Product of Indices/Semigroup

   $\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$

5. Finite Semigroup Equal Elements for Different Powers

   $\forall x \in S: \exists m, n \in \N: m \ne n: x^m = x^n$

6. Element has Idempotent Power in Finite Semigroup

   $\forall x \in S: \exists i \in \N: x^i = x^i \circ x^i$

7. Inverse of Product/Monoid/General Result

   $\forall n \in \N_{> 0}: \left({a_1 \circ a_2 \circ \cdots \circ a_n}\right)^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$

8. Index Laws for Monoids/Negative Index
9. Powers of Group Elements/Negative Index
10. Powers of Group Elements/Sum of Indices
11. Powers of Group Elements/Product of Indices
12. Multiple of Ring Product