Ques. 12
If Y = {0, 1, 2, 3, 4} is transversal for 5Z
in (Z, +). Show whether or not Y is a
subgroup of 5Z
subgroup under addition of integers modulo
of 5
Ques. 11
Let {H_α } ∈ Ω be a family of subgroup of
a group G then prove that ∩_(α=Ω) H_α is also a
subgroup
Ques. 12
Using GAP, find the elements A, B and
C in D_5 such that AB = BC but A ≠ C.
Ques. 8
Find the signum (sign or sgn) of the
permutation θ=(12345678).
Hint : for any permutation β, take
sgn β = {_(−1 if β is odd) ^(1 if β is even)
Ques. 9
Prove that ∣S_n ∣ = n!.
Ques. 10
Provd that for b∈S_n , sgn b = sgn b^(−1) .
Let n & k be positive integers and let
S be a set of n points in The plane such that :
For any point P of S there are at least K points of S Equidistant from p
Prove that k<(1/2)+(√(2n))
A(-1, 2), B(3, 5) and C(4, 8) are the vertices of triangle ABC. Forces whose magnitudes are 5N and 3√10N act along (AB) ⃗ and (CB) ⃗ respectively. Find the direction of the resultant of the forces.
Ques. 6
Let (G, ∗) be a group. and let
C={c∈G : c∗a = a∗c ∀a∈G}. Prove
that C is subgroup of G. hence or
otherwise show that C is Abelian.
[Note C is called the center of group G]
Ques. 7
If (G, ∗) is a group such that (a∗b)^2
= a^2 ∗b^2 (multiplicatively) for all
a,b∈G. Show that G must be Abelian
Let G be a finite group,f be an automorphism of G
such that f(x)=x ⇒x=e .
Then prove that,
(i)∀g∈G, ∃x∈G such that g=x^(−1) f(x).
(ii)If ∀x∈G , f(f(x))=x ⇒ G is Abelian.