Algebra

I'm actually so ashamed of keep asking Weicang on Algebra definitions that i thought i better type them out in an effort to remember them.

Group G
a set of elements with a binary operation defined on it such that x.y=z, where x,y,z are elements in G, . is the well defined binary op.
for every group there exist an identity element such that x.1=1.x=x
the binary op is associative
for every element in the group there exist an inverse such that x.y=1, note that the op may not be commutative

examples of groups are: even numbers, prime number congruent modulo group, 2x2 matrices

Homomorphism H
a mapping on a group such that H(x.y)=H(x)H(y), note that x.y remains as an element in G

Isomorphism I
a bijective homomorphism

Automorphism
A isomorphism from a group to itself, domain = codomain
thus preserving the structure of the group
the set of all automorphism of an object itself is a group
the automorphism group

More important for me...
In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
And once you take all possible permutations and generalize them, you can an automorphism group.
and when two groups are representing two different graphs are actually isomorphic, they share the same structure and problems under one graph can be carried over and study on then.