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Groups

 
 

Group Conditions

A #~{group}, ~G, is a semigroup with an identity element, and in which every element has an inverse. To recap:

~G is a #~{multiplicative group} if:

  1. ( ~x ~y ) ~z _ = _ ~x ( ~y ~z ) , _ &forall. ~x, ~y, ~z &in. ~G
  2. &exist. 1 such that _ 1 ~x _ = _ ~x 1 _ = _ ~x , _ for any ~x &in. ~G
  3. &forall. ~x &in. ~G , &exist. ~x^{-1} &in. ~G such that _ ~x ~x^{-1} _ = _ ~x^{-1} ~x _ = _ 1

~G is an #~{additive group} if:

  1. ( ~x + ~y ) + ~z _ = _ ~x + ( ~y + ~z ) , _ &forall. ~x, ~y, ~z &in. ~G
  2. &exist. 0 such that _ 0 + ~x _ = _ ~x + 0 _ = _ ~x , _ for any ~x &in. ~G
  3. &forall. ~x &in. ~G , &exist. -~x &in. ~G such that _ ~x + -~x _ = _ -~x + ~x _ = _ 0

[These two definitions can be seen to be exactly equivalent, we really only need to differentiate between the two operations in structures that have both, i.e. .]

Group Properties

A group ~G has the following properties (using multiplicative notation and for _ ~a, ~b, ~x &in. ~G ):

  1. The identity element is unique _ (as shown for a groupoid)
  2. The inverse of every element is unique _ (~{ditto})
  3. ( ~x^{-1} )^{-1} _ = _ ~x _ (~{ditto})
  4. ( ~x ~y )^{-1} _ = _ ~y^{-1} ~x^{-1} , _ [ ( ~x ~y ) ( ~y^{-1} ~x^{-1} ) = ~x ( ~y ~y^{-1} ) ~x^{-1} = ~x ~x^{-1} = 1 ]
  5. ~a ~x = ~b ~x _ => _ ~a = ~b , _ [ ~a ~x = ~b ~x _ => _ ~a ~x ~x^{-1} = ~b ~x ~x^{-1} _ => _ ~a = ~b ]
  6. ~x ~x = ~x _ => _ ~x = 1 , _ [ ~x = ~x ( ~x ~x^{-1} ) = ( ~x ~x ) ~x^{-1} = ~x ~x^{-1} = 1 ]

 

Subgroup

A subset _ ~H &subset. ~G , _ is a #~{subgroup} of ~G , if it is a group in its own right, i.e.
_ _ _ _ ~x , ~y &in. ~H _ => _ ~x~y &in. ~H ; _ ~x &in. ~H _ => _ ~x^{-1} &in. ~H _ ( => _ 1 &in. ~H )
Write _ ~H < ~G _ if ~H is a subgroup of ~G.

A sufficient condition for ~H being a subgroup is that, _ &forall. ~x , ~y &in. ~H , _ ~x ~y^{-1} &in. ~H
If condition true then _ ~x ~x^{-1} = 1 &in. ~H . _ So _ 1 ~y^{-1} = ~y^{-1} &in. ~H _ (similarly ~x^{-1} &in. ~H) , _ and so _ ~x ( ~y^{-1})^{-1} = ~x ~y^ &in. ~H

Powers

In a group ~G, for ~n &in. &naturals. define _ ~x^~n _ #:= _ ~x~x ... ~x _ ( ~n times ). _ Obviously, _ ~n , ~m &in. &naturals. _ then _ ~x^~n~x^~m = ~x^{~n + ~m} .
By convention we put ~x^0 = 1 , _ so ~x^~n~x^0 = ~x^~n , so we now have _ ~x^~n~x^~m = ~x^{~n + ~m} , _ &forall. ~n , ~m &in. #Z^+.
Now define _ ~x^{-~n} #:= (~x^{-1})^~n , _ so _ ~x^{-~n}~x^~m = (~x^{-1})^{~n}~x^~m = (~x^{-1})^{~n - 1} ~x^{-1}~x^~m = ~x^{-(~n - 1)}~x^{~m - 1}.
Continuing in the same fashion it is easily seen that if _ ~m > ~n _ then _ ~x^{-~n}~x^~m = ~x^{~m - ~n} , _ if _ ~n > ~m _ then _ ~x^{-~n}~x^~m = ~x^{-(~n - ~m)} = ~x^{~m - ~n} , _ and if _ ~m = ~n _ then _ ~x^{-~n}~x^~n = ~x^{~n - ~n} = ~x^0 = 1 .
Exactly analgous arguments apply to _ ~x^~m~x^{-~n}. _ Finally _ ~x^{-~n}~x^{-~m} = (~x^{-1})^{~n}(~x^{-1})^{~m} = (~x^{-1})^{(~n + ~m)} = ~x^{-~n + -~m} .

So _ ~x^~n~x^~m = ~x^{~n + ~m} , _ &forall. ~n , ~m &in. #Z.

Order

In a group, the ~#{order} of an element is the smallest ~n &in. &naturals. such that ~x^~n = 1 , if this exists. _ Write _ o( ~x ) = ~n . _ If no such ~n exists then ~x is said to have infinite order , _ o( ~x ) = &infty.

~G is a #~{finite} group if it has only a finite number of elements. The number of elements of a finite group is called the ~#{order} of the group, written _ | ~G | .

In a finite group, ~G , the order of any element divides the order of ~G :
Let ~g be any element of ~G , ­ | ~G | = ~n, _ and let o( ~g ) = ~m . _ Then _ ~m =< ~n , _ since \{ 1, ~g , ~g^2 , ... ~g ^{~m - 1} \} are ~m distinct elements of ~G , which has ~n elements.
Now define the relationship: _ ~x ~= ~y _ <=> _ ~x = ~g ^~k ~y , _ for some ~k &in. &naturals. . _ This is an equivalence relation, since _ ~x = ~g ^~m ~x ; _ ~y = ~g^{-~k} ~x = ~g ^{~j~m - ~k} ~x , _ any ~j &in. &naturals. _ (just chose ~j large enough for _ ~j~m > ~k ) ; _ and if _ ~y = ~g ^~p ~z , _ then _ ~x = ~g ^~k ~g ^~p ~z = ~g ^{~k + ~p} ~z . _ Let [ ~x ] be the equivalence class of ~x , i.e. \{ ~x , ~g~x , ~g^2~x , ... , ~g^{~m - 1}~x \}. _ This has ~m elements, as has every other such equivalence class. _ Now _ ~G _ = _ &union._{~x &in. ~G} [ ~x ] , _ so if there are ~q distinct equivalence classes then _ ~n _ = _ ~m ~q .

Corollary: _ ~x ^~n _ = _ 1 , _ &forall. ~x &in. ~G .

 

Multiplication Table

In a finite group ~G of order ~n , _ with elements 1 , ~x_1 , ... , ~x_{~n - 1} , _ the group operation can be defined explicitly in the form of a table (a "multiplication table" if multiplictaive notation is used) which has the following structure:

array{ , #{1} , #{~x_1} , #{~x_2} , ... , #{~x_{~n - 1}}/ #{1} , 1 , ~x_1 , ~x_2 , ... , ~x_{~n - 1}/ #{~x_1} , ~x_1 , ~x_1~x_1 , ~x_1~x_2 , ... , ~x_1~x_{~n - 1}/ #{~x_2} , ~x_2 , ~x_2~x_1 , ~x_2~x_2 , ... , ~x_2~x_{~n - 1}/ ... , ... , ... , ... , ... , ... / #{~x_{~n - 1}} , ~x_{~n - 1} , ~x_{~n - 1}~x_1 , ~x_{~n - 1}~x_2 , ... , ~x_{~n - 1}~x_{~n - 1}/ }

In any group ~G , _ ~a~x = ~a~y _ => _ ~a^{-1}~a~x = ~a^{-1}~a~y _ => _ ~x = ~y . _ For any _ ~a &in. ~G _ define _ ~f_~a #: ~G -> ~G _ by _ ~f_~a ~x = ~a ~x . _ Then ~f_~a is injective (by above) and surjective and is therefore a permutation of ~G.

I.e. each row (or column) of the multiplication table of ~G is a permutation of the elements of ~G.

 

 

Homomorphism

Let ~G be a group and ~H a groupoid. _ A mapping _ ~f #: ~G -> ~H _ is a #~{(group) homomorphism} if

  • ~f ( ~x ~y ) _ = _ ~f ( ~x ) ~f ( ~y ) , _ &forall. ~x , ~y &in. ~G .

Image

If _ ~f _ is a homomorphism, then the #~{image} of ~f , _ Im ~f _ = _ ~f ( ~G ) _ = _ \{ ~h &in. ~H | ~h = ~f ( ~g ) , some ~g &in. ~G \}

a) _ Im ~f _ is a group .

  • closed: _ ~f ( ~x ) ~f ( ~y ) _ = _ ~f ( ~x ~y ) _ &in. Im ~f
  • associative: _ ~f ( ~x ) ( ~f ( ~y ) ~f ( ~z ) ) _ = _ ~f ( ~x ) ~f ( ~y ~z ) _ = _ ~f ( ~x ~y ~z ) _ = _ ~f ( ~x ~y ) ~f ( ~z ) _ = _ ( ~f ( ~x ) ~f ( ~y ) ) ~f ( ~z )
  • identity: _ ~f ( ~x ) ~f ( 1_~G ) _ = _ ~f ( ~x 1_~G ) _ = _ ~f ( ~x ) _ = _ ~f ( 1_~G ) ~f ( ~x ) , _ _ &therefore. _ 1_{Im ~f} _ = _ ~f ( 1_~G )
  • inverse: _ ~f ( ~x ) ~f ( ~x^{-1} ) _ = _ ~f ( ~x ~x^{-1} ) _ = _ ~f ( 1_~G ) _ = _ 1_{Im ~f} , _ similarly _ ~f ( ~x^{-1} ) ~f ( ~x ) _ = _ 1_{Im ~f} , _ _ so _ ~f ( ~x^{-1} ) _ = _ ( ~f ( ~x ) )^{-1}

b) _ If &exist. 1_~H &in. ~H _ then _ ~f ( 1_~G ) = 1_~H .

  • 1_~H identity in ~H , _ then if _ ~f ( ~x ) &in. Im ~f , _ ~f ( ~x ) 1_~H _ = _ 1_~H ~f ( ~x ) _ = _ ~f ( ~x ) , but _ Im ~f _ is a group with identity 1_{Im ~f} _ = _ ~f ( 1_~G ), _ so _ ~f ( 1_~G ) _ = _ 1_~H

Kernel

If _ ~f #: ~G -> ~H _ is a (group) homomorphism, and ~H is itself a group, with identity _ 1_~H , _ then the #~{kernel} of ~f , _ ker ~f _ = _ \{ ~x &in. ~G | ~f ( ~x ) = 1_~H \}

#{Theorem}: _ ker ~f _ < _ ~G , _ for suppose _ ~x , ~y &in. ker ~f , _ then
~f ( ~x ~y^{-1} ) _ = _ ~f ( ~x ) ~f ( ~y^{-1} ) _ = _ ~f ( ~x ) ( ~f ( ~y ) )^{-1} _ = _ 1_~H 1_~H^{-1} _ = _ 1_~H , _ _ so _ ~x ~y^{-1} _ &in. _ ker ~f .

Isomorphism

A bijective homomorphism _ &phi. #: ~G -> ~H _ is called an #~{isomorphism}. _ If such a mapping exists then ~G and ~H are said to be #~{isomorphic}.

Cyclic Group

A ~#{cyclic group} is one in which every element is expressible as the power of one given element called the #~{generator}. _ I.e. _ ~G _ is cyclic if _ &exist. ~g &in. ~G _ such that _ for any ~x &in. ~G , _ ~x = ~g^~n , for some ~n &in. #Z

If ~G is a finite cyclic group of order ~n , write ~G = ~C_~n = \{ 1 , ~c , ~c^2 , ... , ~c^{~n - 1} | ~c^~n = 1 \} . _ ~G is cyclic of order ~n _ <=> _ &exist. ~c &in. ~G , _ o ( ~c ) = ~n .

In additive notation a cyclic group of order ~n is written: _ _ _ ~G_~n _ = _ \{ 0, ~g , 2~g , ... , ( ~n - 1 ) ~g | ~n~g = 0 \} _ _ [ Here _ ~k~g _ denotes _ ~g + ~g + ... + ~g _ ( ~k times )] .

Two cyclic groups of order ~n are isomorphic: _ ~A_~n _ = _ \{ 1 , ~a , ~a^2 , ... , ~a^{~n - 1} \} , _ ~B_~n _ = _ \{ 1 , ~b , ~b^2 , ... , ~b^{~n - 1} \} , _ take _ ~f #: ~A_~n -> ~B_~n , _ such that _ ~f ( ~a ^~i ) = ~b ^~i , _ ~i = 0, 1, ... , ~n - 1 .

Any cyclic group is abelian, for _ ~a ^~i ~a ^~j _ = _ ~a ^{~i + ~j} _ = _ ~a ^~j ~a ^~i .