Houjun Liu



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

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