Abstract Algebra Answers

1. Suppose G is an abelian group of order 6 containing an element of order 3. Prove G is cyclic?
2. Suppose G has only one element (say a) of order 2. show xa=ax for all x in G?

Prove that the group G=[a,b] with the defining set of relations a³=e, b⁷=e, a^-1ba=b⁸ is a cyclic group of order 3?

Use the Fundamental Theorem of Homomorphism for Groups to prove the following 

theorem, which is called the Zassenhaus (Butterfly) Lemma: 

Let H and K be subgroups of a group G and H′ and K′ be normal subgroups of H 

and K, respectively. Then 

 i) H′(H ∩ K′) H′(H ∩ K)

H (H K)

′ ∩ ∩ ′

∩ − ′ ′∩

′ ∩ − ′ ∩ ′

′ ∩ (15)

The situation can be represented by the subgroup diagram below, which explains the

name ‘butterfly’

consider the dihedral group D₆ and define its action on X={1,2,3,4,5,6} ?

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, what is the permutation group to which a cyclic group of order 12 is isomorphic to?

Let G be a subgroup of GL₂ (Z₄) defined by the set {[m b ,0 1}] such that b ∈ Z₄ and m=±1. Show that G is isomorphic to a known group of order 8?

check whether or not Q[x]/<8x^3+6x^2-9x+24> is a feild or not

Find the cyclic subgroups of U(21).

Prove that (Z(sqrt(2)),+,×) is an integral domain.

Subject: Rings and fields