Abstract Algebra

Assignment #1

Kenneth Levasseur
Mathematical Sciences
UMass Lowell
Kenneth_Levasseur@uml.edu

Due  Sept. 26.

Instructions: Use only the definition of a group and the definition of exponentiation in groups in doing these proofs.

Definition of exponentiation in a group:  If g ∈G and n is a positive integer,
		(g^0)= e
(g^n)= (g^(n-1))*g
(g^(-n))=((((g^n)))^(-1))

You may also assume that for all positive n, (g^n)= g*(g^(n-1)).   This isn’t true from the definition, but can be proven by induction.

1.  Prove that if n is a positive integer, then (g^(-n))=((( (g^(-1))))^n).   In other words, in computing (g^(-n)) you can invert g and then raise the result to the (n^(th ))power instead of raising g to the (n^th)power and then inverting the result.  You get the same result either way.

2. Let a and b be elements in a group G. Prove that a (b^n)(a^(-1)) = (((a b (a^(-1))))^n) for all n≥1.

3.  Let G be a finite group with identity e.   Prove that if a∈G, there exists a positive integer m such that (a^m)= e.

Exploring Abstract Algebra with Mathematica

Home Page:  http://www.central.edu/eaam/index.asp

You can download the AbstractAlgebra package from the site above, but you can also load it directly with this input:

Import["http://homepage.mac.com/klevasseur/Master.m"]

A few simple inputs.

Z[5]

CayleyTable[Z[5]]

Dihedral[3]

Dihedral[3,Mode->Visual]

SubgroupGenerated[Dihedral[3],Ref]