1. Prove that a cycle of length l is odd if l is even.
Prove that the product of an even permutation and an odd permutation is odd.
— If m1, m2 belongs to the same orbit then St(m1) and St(m2) are conjugate to each other if m2=p(g)m1 then St(m2)=gSt(m1) g ...
Find a maximal ideal of R [x] containing
the ideal <x²— 1, x³— 1>
Which of the following statements are true, and which short proof or a counter-example.
i)There is no non-abelian group of 12
ii)If in a group G every element is of finite order, then G is a finite order.
iii)The homomorphic image of a non-cyclic group is non cyclic.
iv) If a is an integral domain, then R /I is an integral domain for every non-zero ideal I of R .
v)If I and J are ideals of a ring R, then so is I U J