Suggest edit — group

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Note

---
title: "group"
type: concept
related: [Operation, Commutivity, Inverses, Group, Objects]
source: https://www.jemoka.com/posts/kbhgroup/
confidence: high
status: active
---

components a set of constituent objects an operation requirements for group closed: if \(a,b \in G\), then \(a \cdot b \in G\) existence of identity: there is \(e \in G\) such that \(e\cdot a= a\cdot e = a\), for all \(a \in G\) existence of inverses: there is \(b \in G\) for all \(a \in G\) such that \(a\cdot b = b\cdot a = e\) associative: \((a\cdot b)\cdot c = a\cdot (b\cdot c)\) for all \(a,b,c \in G\) additional information identity in group commutates with everything (which is the only commutattion in groups
Unique identities and inverses the identity is unique in a group (similar idea as additive identity is unique in a vector space) for each \(a \in G\), its inverse in unique (similar ideas as additive inverse is unique in a vector space) cancellation policies if \(a,b,c \in G\), \(ab = ac \implies b = c\) (left cancellation)
\(ba = ca \implies b = c\) (right cancellation)
sock-shoes property if \(a,b \in G\), then \((ab)^{-1} = b^{-1}a^{-1}\)