(G,o) be a semigroup & a,b in G, if the equations aox=b, yoa=b has a root in G then (G,o) is a group

G be a finite group and H◁G then order of an element gH in G/H is a divisor of the order of g in G

If G be a non-commutative group with center Z , then the quotient group G/Z is non-cyclic.

Centraliser of an element 'a' of a group G , C(a) is a sub-group of the group.

Centre of a group G , Z(G) - is a sub-group of the group G .

If H be a subgroup of a cyclic group G , then the quotient group G/H is cyclic.

Quotient Group

Let H be a subgroup of G. Then H is normal in G iff for h∈H and x∈G ⇒ xhx⁻¹∈H .

The set of all distinct left cosets and right cosets have the same cardinality.

G be a cyclic group of order n.Then for every+ve divisor m of n,G has a unique subgroup H of order m

For every positive integer n , ∑φ(d)=n ; where the sum being taken for all positive divisors d of n

Lagrange's Theorem:The order of every subgroups of a finite group G, is a divisor of the order of G

Every subgroups of a cyclic group is cyclic.

RSA-scheme.

A function f which is continuous on a closed interval [a,b] is also uniformly continuous on [a,b]

Function f is continuous on [a,b] and f(a) ≠f(b) , then it assumes every value between f(a) and f(b)

f is continuous on [a,b] and f(a).f(b) is ( -ve) then ∃ at least one point α∈(a,b) such that f(α)=0

An important property of continuous functions defined on a closed interval [a,b] .

f is continuous on a closed interval [a,b] , then it attains it's bounds at least once in [a,b] .

If a function f is continuous on a closed interval [a,b] , then it is bounded therein.

f on I is continuous at a point c ∈ I iff for every sequence {cₙ} in I , cₙ→c , we have, f(cₙ)→f(c)

Proof of Cauchy's Criterion for finite limits

The derived set S' of a bounded infinite set S (⊂R) has the smallest and the greatest members.

The derived set of a bounded set is bounded.

The supremum(infimum) of a bounded nonempty set S(⊂ R),when not a member of S,is a limit point of S

A closed set either contains an interval or else is nowhere dense.

The derived set of a set is closed.

Union of an arbitrary family of open sets is open and intersection for closed sets is closed .

A set is closed iff its complement is open.

Bolzano-Weierstrass Theorem: Every infinite bounded set has a limit point.

Definition of limit point.

Interior of S i.e. int(S)=S⁰ - is an open set , moreover it is the largest open set contained in S.

Every open interval (a , b) contains a rational number .

Archimedean Property in R .

Dedekind's property and Order-Completeness property in R are equivalent.

Show that set of all rational numbers is not order-complete .

Show that √8 is not a rational number .

UPSC , SSC CGL exam ❓

Find the whole length of the loop of the curve 9ay²=(x-2a)(x-5a)².

Find The Equation of The Cone obtained by revolving the straight line x-2y=1 about X-axis .