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Isomorphism
a one-to-one matching (dictionary) between two groups members that converts each true equation in one group into true equation in other
Subgroup
(self contained group on its own) if 1) Identity - collection includes identity, 2) Combinations - when any pair of members of collect are combined, answer lies in collection, 3) Inverses - inverse of anything in the collection lies in the collection
Even permutation
permutation that can be obtained by composing an even number of swaps
Odd permutation
permutation that can be obtained by composing an odd number of swaps
Nth Alternating group
subgroup of all even permutations in Pn is denoted An
One-to-one manner
Every permutation must correspond to one and only one symmetry
Fully rigidly equivalent
if there exists rigid motion of space which, when applied to first object, repositions it so that afterwards, the 2 objects have exactly same symmetries
Properly rigidly equivalent
if there exists rigid motion of space, which when applied to first object, repositions it so that afterwards 2 objects have exactly same proper symmetries
Essentially 2-D
if 3D object (not counting axes of order 2) has 0 or 1 rotation axis
Chiral
if all of 3D objects symmetries are proper
Central inversion
reflecting through a point
Centrally symmetric
object is this if central inversion (across its center point) is a symmetry of the object
