## components

## 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 \((ab)^{-1} = b^{-1}a^{-1}\)