Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 24 September 2014

Lecture 21: Cyclic groups and products

The general linear group with entries in is GLn() = {n×nmatricesgwith entries insuch thatgis invertible} = { gMn()| g-1Mn() } = { gMn()| det(g)0 } . So GL2()= { (abcd) |a,b,c,d andad-bc0 } with product matrix multiplication. So GL1() = {gM1()|det(g)0} = {c|c0} = -{0} = ×.

A cyclic group is a group generated by one element.

3={0,1,2} is generated byS={1}. {,,} is generated byS={}. {,,,} is generated byS={}. {,,,,} is generated byS={}. {1,g,g2,g3,g4} withg5=1 is generated byg. In this last example g3g4=g7= g5g2=1· g2=g2.

{1,-1+3i2,-1-3i2} (a subset of ) is a group under multiplication. If ζ=-1+3i2 then ζ2=-1-3i2 and ζ3=1 so that {1,-1+3i2,-1-3i2} ={1,ζ,ζ2}with ζ3=1 and this group is generated by S={ζ}={-1+3i2}.

The group of nth roots of unity is μn={z|zn=1}. The group of 3rd roots of unity is {1,-1+3i2,-1-3i2}=μ3. Then μn={z|zn=1} is a subgroup of GL1()=× and μn is generated by η=e2πi/n where e2πi/n= cos(2πn)+ isin(2πn). Note: e2πi/3= cos(2π3)+ isin(2π3)= -12+i32= -1+3i2.

Let G and H be groups. The product of G and H is the set G×H={(g,h)|gGandhH} with (g1,h1) (g2,h2)= (g1g2,h1h2).

2={0,1} and 2×2 ={(0,0),(0,1),(1,0),(1,1)} with (0,0) (0,1) (1,0) (1,1) (0,0) (0,0) (0,1) (1,0) (1,1) (0,1) (0,1) (0,0) (1,1) (1,0) (1,0) (1,0) (1,1) (0,0) (0,1) (1,1) (1,1) (1,0) (0,1) (0,0) Order of2×2is4 Order of(0,0)is1 Order of(1,0)is2 Order of(0,1)is2 Order of(1,1)is2

{,,,} is a subgroup of S4. Order of{,,,}is4 Order ofis1 Order ofis2 Order ofis2 Order ofis2 and the function 2×2 {,,,} (0,0) (1,0) (0,1) (1,1) is an isomorphism.

Subgroups of 2×2: 2×2 {(0,0),(1,0)} {(0,0),(0,1)} {(0,0),(1,1)} {(0,0)} Subgroups of {,,,}: {,,,} {,} {,} {,} {}

Notes and References

These are a typed copy of Lecture 21 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on September 9, 2011.

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