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

In this section we will study on multiplication(Product) of vectors.

PRODUCT OF TWO VECTORS:-

this is of two types

i)Scalar Product

ii)Vector Product

=>>Scalar Product-

When product of two vectors is a scalar quantity,it is called ‘Scalar Product’.

If a & b be two vectors then their scalar product is denoted by ‘a.b’ and

a.b=|a||b|cos@ or

a.b=lmcos@

(`.`|a|=l etc.)

Important for a.b

i)a.b=b.a

ii)a.a=|a|2=l2

iii)a.b=0 =>a¿b

iv)i.i=j.j=k.k=1

i.j=j.k=k.i=0

v)if a=pi+qj+rk and

b=si+tj+uk ,

then

a.b=ps+qt+ru

vi)cos@=a.b/lm

=>>Vector Product-

when product is a vector quantity,it is called ‘Vector Product’.

It is written as ‘a*b’ and

a*b=|a||b|sin@ñ

a*b=lmsin@ñ

where ñ is u.v. along

a*b

Important for a*b

i)a*b=-b*a i.e. a*b is not equal to b*a

ii)a*a=0

iii)a*b=0 =>a || b

iv)i*i=j*j=k*k=0 ,and

i*j=k

j*k=i

k*i=j

v)a*b=

[i j k]

|p q r|

[s t u]

vi)if a & b are adjacent sides of ||gm, area of ||gm= |a*b|, and if sides of triangle then area=half of |a*b| i.e. Area=|a*b|/2

vii)if a and b are diagonals of ||gm ,area=|a*b|/2

viii)sin@=|a*b|/lm

This equation is generally written as aX2+2hXY+bY2=0 __(i)

Now we know that a line through origin is y=mx, let two represented by (i) be
y=m1x and
y=m2x
then
aX2+2hXY+bY2=b(y-m1)(y=m2x)
this gives
m1+m2=-2h/b and m1.m2=a/b

Also angle between (ii) is @ then
tan@=m1~m2/1+m1.m2
tan@=(h2-ab)1/2/(a+b)

Let we have to solve the following system
x+y+z=3
2x-y+z=2
x-2y+3z=2 …..(i)
this system can be put in the form “AX=B” or “A=X-1B” ….(ii)
where A=
[1 1 1]
|2 -1 1|
[1 -2 3]
X=
[x]
|y|
[z]
B=
[3]
|2|
[2]
now |A|=-9

Now we are to find ‘adjA’ and it
adjA=
[A11 A12 A13] [-1-5-3]
|A21 A22 A23|=|-5 2 3|
[A31 A32 A33] [-3 1 -3]
hence
A-1=adjA/|A|
and hence
X=A-1B
[x] [-1-5 -3][2]
|y|=-1/9|-5 2 3||2| =
[z] [ -3 1 -3][3]
[x] [-9]
|y|=-1/9|-9|
[z] [-9]
This gives
x=1, y=1, z=1

QUANTITIES and their REPRESENTATION:
These are two in number- ¡)Scalar ¡¡)Vector
Scalar has only magnitude but Vector has magnitude as well as direction.A Vector is denoted generally by english alphabet a b c with arrow or cap on its head as ‘å’ while scalar by x y z k l m n without arrow.
Here we will denote a Vector by ‘a b c’ and unit vectors by å etc.Magnitude of a b c by l m n and unit vector along axes by i j k while unit vector along normal by ñ.

ADDITION OF VECTOR:-
Let two vectors be a & b as shown in fig_1(a) and R be the resultant of a & b,then R is given by R=a+b and its direction will be as shown in fig_1(b).

SUBSTRACTION OF VECTORS:-
Let R’=a-b
R’=a+(-b)
see in fig_1(c) i.e. the direction of ‘b’ is changed.

An equation in which the sum of powers all variables in each term is same is called homogeneous equation. If this sum is ‘n’ then we say that equation is homogeneous of n th degree. E.g.
1>aX+bY=0
This is homo. of 1st degree.
2>aX2+2hXY+bY2=0
This is of 2nd degree.