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}