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.
 * 1) 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.
 * 1) Index Laws/Sum of Indices/Semigroup
 * $\forall m, n \in \N_{>0}: a^{n + m} = a^n \circ a^m$
 * 1) Index Laws/Product of Indices/Semigroup
 * $\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$
 * 1) Finite Semigroup Equal Elements for Different Powers
 * $\forall x \in S: \exists m, n \in \N: m \ne n: x^m = x^n$
 * 1) Element has Idempotent Power in Finite Semigroup
 * $\forall x \in S: \exists i \in \N: x^i = x^i \circ x^i$
 * 1) Inverse of Product/Monoid/General Result
 * $\forall n \in \N_{> 0}: \left({a_1 \circ a_2 \circ \cdots \circ a_n}\right)^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$
 * 1) Index Laws for Monoids/Negative Index
 * 2) Powers of Group Elements/Negative Index
 * 3) Powers of Group Elements/Sum of Indices
 * 4) Powers of Group Elements/Product of Indices
 * 5) Multiple of Ring Product