Definition·D046
Group isomorphism
A bijective homomorphism; isomorphic groups are the same group up to renaming elements.
An isomorphism of groups is a homomorphism
that is also a bijection. Groups
and
are isomorphic, written
, when
.
In words
An isomorphism is a homomorphism from G to H that is also a bijection. Groups G and H are isomorphic when such an isomorphism exists.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
The inverse of a group isomorphism is again an isomorphism (it is a bijection, and it preserves the operation because
does), so isomorphism is symmetric; with the identity and composition it is reflexive and transitive, an equivalence relation on any set of groups. Isomorphic groups share every group-theoretic property: order, commutativity, the subgroup structure, and all element orders. By T16 any two groups of the same prime order are isomorphic.