Question

1. a) State and prove Ca
b). Prove that the set I(G) of all inner automorphisms of a group G is a normal 3
subgroup of A(G), the group of all automorphisms of G and that I(G) is
isomorphic to G/Z, where Z is the centre of G.
State and prove Cauchy's theorem for finite groups. 46
(4 + 5 + 5)
2. a State and prove Sylow's theorem.So
b) If G is a group of order 231, show that 11 - Sylow subgroup of G is contained in
the centre of GoSyl0w 30 thootine i 155)
c) Define a solvable group. If G is a group and H is a subgroup of G; then show that
H is solvable. Based on problem (58)
(4 +6+4)
3. a Tf in a ring R, x = x for all x then show that 2x = 0 and x + y = 0 imply x=y.
b) Prove that the ideal S of the ring I of all integers is maximal if and only if S is
generated by some prime integer.
refer pare
c) Find the units in the ring of Gaussian integers and show that they form a
multiplicative abelian group. refer Page 15
(3 +4+7)​

Answer 1

Answer:

Rinse the cut or wound with water and apply pressure with sterile gauze, a bandage, or a clean cloth. If blood soaks through the bandage, place another bandage on top of the first and keep applying pressure. Raise the injured body part to slow bleeding. When bleeding stops, cover the wound with a new, clean bandage.

Step-by-step explanation: