Suppose a group G with operation &comp. has a non-empty subset H, _ then H is a #~{subgroup} of G if, for every ~x, ~y &in. H ,
~x &comp. ~y &in. H , _ and _ ~x^{-1} &in. H .
Thus H is a group in its own right under the operation &comp. restricted to H.
If H is a subgroup of G, then the identity element of G is in H, and is the identity elelment of H as a subgroup.
#{Example}
The group _ G = \{ &iota. , &alpha. , &beta. , &gamma. , &delta. , &epsilon. \} _ of of a triangle has six subgroups:
\{ &iota. \} ,_ \{ &iota. , &gamma. \} ,_ \{ &iota. , &delta. \} ,_ \{ &iota. , &epsilon. \} ,_ \{ &iota. , &alpha. , &beta. \} ,_ and_ G
Let R be a ring with the operations of addition and multiplication. A non-empty subset T &subseteq. R is called a #~{subring} of R if it is a ring under the operations of R restricted to T.
In fact a subset T of R is a subring if, for all ~x, ~y &in. T,
~x + ~y &in. T , _ -~x &in. T , _ and _ ~x~y &in. T .
There are similar definitions for #~{subfield}, #~{subsemi-group} etc.
If G and H are groups, with operations &comp. and * respectively, then a mapping _ &theta. #: G -> H _ is said to be a #~{homomorphism (of groups)} if
( ~x &comp. ~y ) &theta. _ = _ ~x&theta. * ~y&theta. , _ _ &forall. ~x , ~y &in. G.
#{Example}
Let G be the group of real numbers ( &reals. ) under addition, and let H be the group of positive real numbers ( &reals.^+ ) under multiplication. The mapping _ &theta. #: &reals. -> &reals.^+ , _ ~x&theta. = exp \{ ~x \} _ is a homomorphism, for
( ~x + ~y ) &theta. _ = _ exp \{ ~x + ~y \} _ = _ exp \{ ~x \} exp \{ ~y \} _ = ~x&theta. ~y&theta. , _ _ &forall. ~x , ~y &in. &reals. .
Unless otherwise stated, we will assume that the operation in groups that we consider will be multiplication, in which case the identity element of group G is written 1_G or just 1, the inverse of ~x is written ~x^{-1}, and a homomorphism _ &theta. #: G -> H _ satisfies : _ _
( ~x ~y ) &theta. _ = _ ~x&theta. ~y&theta. , _ _ &forall. ~x , ~y &in. G.
If a group G has a finite number of elements, it is called a #~{finite group} and the number of elements is called the #~{order} of G.
If G is any group (not necessarily finite) and ~x &in. G, then ~x is said to have #~{finite order} _ if _ &exist. ~m &in. &naturals. , ~m > 0 , _ such that _ ~x^~m = 1 . _ The #~{order} of ~x is then the smallest positive ~m such that ~x^~m = 1 . Otherwise ~x is said to have #~{infinite order}.
Lemma: If ~x has order ~m, and _ ~x^~n = 1 , _ some ~n != ~m _ (~n > ~m by definition), _ then ~m divides ~n.
For let _ ~n = ~p~m + ~q , _ ~q < ~m , _ then _ 1 _ = _ ~x^~n _ = _ ~x ^{~p~m + ~q} _ = _ ( ~x^~m ) ^~p ~x^~q _ = _ 1^~p ~x^~q _ = _ ~x^~q _ _ contradiction, unless ~q = 0.
If &theta. #: G -> H _ is a homomorphism (of groups), then
1_G&theta. _ = _ 1_H
( ~x^{-1} )&theta. _ = _ ( ~x&theta. )^{-1}
and if ~x &in. G has finite order, then ~x&theta. has finite order dividing that of ~x
Proof:
~x &in. G , _ ~x&theta. _ = _ ( 1_G ~x ) &theta. _ = _ 1_G&theta. ~x&theta. _ _ => _ _ 1_G&theta. _ = _ 1_H , _ by uniqueness of identity.
~x&theta.( ~x^{-1} )&theta. _ = _ ( ~x ~x^{-1} )&theta. _ = _ 1_G &theta. _ = _ 1_H .
If &exist. ~m > 0 &in. &naturals. _ such that _ ~x^~m _ = _ 1_G , _ then _ (~x&theta.)^~m _ = _ (~x^~m )&theta. _ = _ 1_G &theta. _ = _ 1_H . _ The result follows from the preceeding lemma.
A bijective homomorphism _ &theta. #: G -> H _ is called an #~{isomorphism}. If such an isomorphism exists, G and H are said to be #~{isomorphic}.
If _ &theta. #: G -> H , _ &phi. #: H -> K _ are homomorphisms (or isomorphisms) _ then _ &theta.&phi. #: G -> K _ is a homomorphism (or an isomorphism).
If _ &theta. #: G -> H _ is an isomorphisms then _ &theta.^{-1} #: H -> G _ is an isomorphism.
Proof:
( ~x~y ) &theta.&phi. _ = _ ( ~x&theta. ~y&theta. )&phi. _ = _ ~x&theta.&phi. ~y&theta.&phi. , _ _ &forall. ~x , ~y &in. G.
~w , ~z &in. H then &exist. ~x , ~y &in. G _ such that _ ~w = ~x&theta. , _ ~z = ~y&theta., _ i.e. _ ~x = ~w&theta.^{-1} , _ ~y = ~z&theta.^{-1}, _ then _
( ~w~z )&theta.^{-1} _ = _ ( ~x&theta.~y&theta. )&theta.^{-1} _ = _ ( ~x~y ) &theta.&theta.^{-1} _ = _ ~x~y _ = _ ~w&theta.^{-1}~z&theta.^{-1}
It follows from this theorem that the relation "~{is isomorphic to}" is an equivalence relation on the class of all groups.
If R , S are rings/fields with operations of addition and multiplication, then a mapping _ &theta. #: R -> S _ is called a #~{homomorphism (of rings/fields)} (or just #~{ring/field homomorphism}) if, &forall. ~x , ~y &in. R :
( ~x + ~y )&theta. _ = _ ~x&theta. + ~y&theta.
( ~x~y )&theta. _ = _ ( ~x&theta. )( ~y&theta. )
If &theta. is bijective then it is called an #~{isomorphism (of rings/fields)}
Regarding R and S as groups under addition, we see that a homomorphism of rings is also a group homomorphism.
As in the theorem on order for groups, it follows that _ 0_R&theta. = 0_S , _ and _ ( -~x )&theta. = - ( ~x&theta. ) _ _ &forall. ~x &in. R.
#{Example}
Let Z_~m denote the ring _ \{ 0, 1,2, ... , ~m - 1 \} _ under addition and multiplication modulo ~m. Then there are two homomorphisms _ &theta._1 , &theta._2 #: Z_4 -> Z_2 ,
0&theta._1 = 0 , _ 1&theta._1 = 0 , _ 2&theta._1 = 0 , _ 3&theta._1 = 0
0&theta._2 = 0 , _ 1&theta._2 = 1 , _ 2&theta._2 = 0 , _ 3&theta._2 = 1
As in the theorem on homomorphisms for groups, if _ &theta. #: R -> S , _ &phi. #: S -> T _ are ring homomorphismes, then _ &theta.&phi. #: R -> T _ is a ring homomorphism, and if &theta. is a ring isomorphism then so is &theta.^{-1}.