A/a art. 1.one, any
2. first letter of english alphabet.

A 1 or A-one first rate, first class, informal excellent.

aback adv. towards the back,
taken~ surprised

abandonvt. give up
abase vt. degrade.
abash vt. embarras, disconcert, put shame.
abate vt.,vi. make or become less; ~ment n. reduction.

abattoirn. a slaughter house.
abbotn. head of community of monks.
abdicatevt. renounce the throne or responsibility.
abductvt. kidnap, carry off illegally.
abedadv. in bed

Def. A non-empty set ‘G’ is called a group w.r.t. the binary operation’o', iff it satisfies the following axioms-
G1)Closure axiom
¥ a, b € G => aob € G.
G2)Associative axiom
¥ a, b, c€ G
ao(boc)= (aob)oc
G3)Identity axiom
if ‘e’ be identity of ‘G’
then aoe=eoa=a, ¥ e€G
G4)Inverse axiom
if ‘p’ be inverse of ‘a’
then aop=poa=e, ¥ a€G.

Abelian group : If G is commutative too,is called ‘abelian group’.
i.e. aob=boa

subgroup :a non-empty subset ‘H’ of a group G is called a subgroup of G is itself a group w.r.t. the binary operation defined on G.
Properties of Subgroups:-
If H is subgroup of G then,
i)H=inverse of H, but converse in not always true.
ii)square of H= HH= H.

Complex :-A non-empty subset ‘H’ of G is called Complex of G.
Properties of Complex:-If H,K,L be three complexes of G, then
i)HK is also a complex.
ii)inverse of H is also a complex.
iii)inverse of (HK)=(inverse of K)*(Inverse of H)

Conditions of H to be a Subgroup :-
1)(i) a, b€ H => ab €H,
(ii)a€ H =>(inverse of a) € H.

2)a, b€ H=>a*(inverse of b) €H.

3)H(inverse of H)=H.

4)a, b€H => ab€H

5)H & K are two subgroup them ‘HK’ is a subgroup iff ‘HK=KH’

A Counting Principle :-If H and K are subgroup, then the subset HK need to be a subgroup of G. We want to determine the number of distinct elements in the subset HK and we denote it by ‘o(HK)’ and
o(HK)=o(H)o(K)/o(HcapK).

Not:coset,modulo,Lagrange’s theorem,

Normalizer:-If G is a group and a€G, then normalizer of a is the set of all those elements of G which commute with a.
Symbolically,Normalizer
N(a) ={xa=ax:x€G}

Normal Subgroup:- A subgroup H is said to be ‘Normal Subgroup’ iff ¥ x€G, and ¥ h€H =>xH(inv of x)€H