Changes

:As another example, we can define four groups as follows

:As another example, we can define four groups as follows

:#The group <math>(S, \cdot)</math>, where <math>S = \{a, b, c, d\}</math> and <math>\cdot</math> is defined according to the table

:#The group <math>(S, \cdot)</math>, where <math>S = \{a, b, c, d\}</math> and <math>\cdot</math> is defined according to the table

{| class="wikitable"

:#:{| class="wikitable"

|-

|-

! \cdot

! \cdot

| <math>c</math>

| <math>c</math>

|}

|}

:#The group <math>(\mathbb{Z}_4, +)</math>, where <math>mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition

:#The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition

:#The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.

:#The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.

:By definition, none of these groups are equal, since groups are technically ordered pairs and none of the underlying sets are equal.  However, each pair of these groups is isomorphic, which means that there exists a bijective homomorphism.  Isomorphic groups have the same group structure.  For example, each of these groups is abelian, which means that the group operation is commutative.  Also, each group has four elements.  Further, each group is cyclic, meaning that it is generated by a single element; the Klein four-group is not cyclic.  Therefore, when we say that there are only two groups of order 4 ''up to isomorphism'', we use isomorphisms and homomorphisms because we really don't care about groups that have the same structure but just have the elements relabeled.

:By definition, none of these groups are equal, since groups are technically ordered pairs and none of the underlying sets are equal.  However, each pair of these groups is isomorphic, which means that there exists a bijective homomorphism.  Isomorphic groups have the same group structure.  For example, each of these groups is abelian, which means that the group operation is commutative.  Also, each group has four elements.  Further, each group is cyclic, meaning that it is generated by a single element; the Klein four-group is not cyclic.  Therefore, when we say that there are only two groups of order 4 ''up to isomorphism'', we use isomorphisms and homomorphisms because we really don't care about groups that have the same structure but just have the elements relabeled.

:Homomorphisms exist between any pair of groups: the most obvious example is the homomorphism mapping every element of the domain group to the identity element of the target group.  Homomorphisms are very interesting for many reasons; the kernel of a homomorphism (the preimage of the identity element) is always a normal subgroup, and free groups <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>.  [[User:GregG|GregG]] 21:16, 28 February 2012 (EST)

:Homomorphisms exist between any pair of groups: the most obvious example is the homomorphism mapping every element of the domain group to the identity element of the target group.  Homomorphisms are very interesting for many reasons; the kernel of a homomorphism (the preimage of the identity element) is always a normal subgroup, and free groups <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>.  [[User:GregG|GregG]] 21:16, 28 February 2012 (EST)