Abstract Algebra Answers

Check whether H={x∈ R^*|x =1or x is irrational}and K={x∈ R^*|x ≥1}are subgroups of (R*,.).

The table below is a Cayley table for the group ({e,a,b,c,d},∗). Fill in the blanks.

e a b c d

_______________________

e e e - - -

a - b - - e

b - c d e -

c - d - a b

d - - - - -

_______________________

Let G be a finite group. Show that the number of elements g of G such that g³= e is

odd, where e is the identity of G.

Give a set of cardinality 5 which is a subset of Z\N.

Give an example, with justification, of a function with domain Z \{2,3}and codomain N.Is this function 1 –1?Is it onto ?Give reasons for your answers.

Give the smallest n ∈ N for which An is non-abelian. Justify your answer.

List two distinct cosets of < r > in , D₁₀ where r is a reflection in . D1

Let τ be a fixed odd permutation in . S₁₀ Show that every odd permutation in S₁₀ is

a product of τ and some permutation in A₁₀

Using Cayley’s theorem, find the permutation group to which a cyclic group of 

order 12 is isomorphic

Prove that a cyclic group with only one generator can have at most 2 elements.