Groups

Notations

e element of

==> implies

" for every

$ there exists

w.r.t with respect to

iff if and only if

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Binary operation

* is said to be a binary operation on a set G (or G is said to be closed under the operation *) if a,b e G ==> ( a*b) e G

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Group

A group < G,*> consists of a nonempty set G and a binary operation * defined on G satisfying the following conditions

(i) Associativity: if a,b,c e G then ( a * b) * c = a* ( b * c )

(ii) Existence of identity: there is an element e e G such that

a * e = a = e * a , for any element a e G

(iii) Existence of inverse: for any a e G , there is an element

a' e G , such that ( a * a' ) = e = ( a' * a ), where e is the identity.

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Abelian group (Commutative group)

A group < G,*> is said to be an abelian group

if a * b = b * a , for every a,b e G

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Subgroup

Let < G,*> be a group. A nonempty subset H of G is said to be a subgroup of G if H itself is a group with respect to the induced operation * .

Note:

Let < G,*> be a group. A nonempty subset H of G is a subgroup of G iff it satisfies the following conditions

(i) a,b e H ==> a * b e H

(ii) a e H ==> ( a^-1 ) e H , where a^-1 represents the inverse of a w.r.t the operation * in G

The two conditions given above can be combined into a single condition given by a,b e H ==> a * b^-1 e H

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Coset

Let < G,*> be a group and H be a subgroup of G. Let a e G . Then the set

given by a*H = { a*h / h e H} is called the left coset of H in G

and H * a = { h*a / h e H} is called the right coset of H in G.

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Normal subgroup

Let < G,*> be a group and H be a subgroup of G. Then H is said to be a normal subgroup if x*h*x^-1 e H , whenever x e G and h e H.

or

Let < G,*> be a group and H be a subgroup of G. Then H is said to be a normal subgroup if x*H = H*x ,whenever x e G and h e H.

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Quotient group

Let < G,*> be a group and H be a normal subgroup of G. Then the collection of cosets, G_/H = {a*H / a e G} is a group under the operation defined by ( a*H) * ( b*H) = (a*b) *H

G_/H is called a quotient group .

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Cyclic group

A group G is said to a cyclic group if there is some element a in G such that

G = {aⁿ/ n e Z}

Note:

(i)aⁿ stands for (a*a*....*a) ntimes.

(ii) in an additive group, na will correspond to aⁿ

(iii) Z represents the set of integers.

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Homomorphism,isomorphism

Let < G,*> and < G', #> be two groups. If f : G ---> G' is a function which satisfies f(a * b) = f(a) # f(b) ," a,b e G , then f is called a homomorphism from G into G' . If f is also one-to-one (injective), then f is called an isomorphism of G into G' . If f : G ---> G' is a homomorphism that is both one-to-one and onto (surjective), then G is said to be isomorphic to G' and this is denoted by G @ G'.

Kernel of a homomorphism

Let G and G' be two groups. If f : G ---> G' is a homomorphism , then

the set ker(f) = {a e G / f(a) = e'}is called the kernel of f.

where e' stands for the identity element of G' .

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