Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 25 September 2014

Lecture 31: Revision: Analogies

Let G be a finite subgroup of Isom(𝔼2). Then G is a cyclic or a dihedral group.


Step 1 Let p𝔼2 and G={g1,,gt}.
Then q=g1p++grp is a fixed point of G.
So every element of G is a rotation about q or a reflection in a line through q.
Let h=rθ,q with θ minimum possible and let H=h.
Then H is a cyclic group.
Case 1:
If H=G then G is cyclic.
Case 2:
If HG let s1,s2G such that s2,s1H and s1s2 then s1s2H so s1Hs2-1=Hs2, since s22=1.
So G={H,s1H} with s12=1.
So G is a dihedral group.


A group is a set G with a function G×G G (g1,g2) g1g2 such that

(a) If g1,g2,g3G then (g1g2)g3=g1(g2g3),
(b) There exists 1G such that if gG then g·1=g and 1·g=g.
(c) If gG then there exists g-1G such that gg-1=1 and g-1g=1.

Homomorphisms are for comparing groups.

A homomorphism from G to H is a function f:GH such that

(a) If g1,g2G then f(g1g2)=f(g1)f(g2).

Let f:GH be a homomorphism. The kernel of f is kerf={gG|f(g=1)} and the image of f is imf={f(g)|gG}.

Examples of groups

Cyclic groups, Dihedral groups, Symmetric groups. Sn = { σ:{1,,n} {1,,n}| σis a bijection } = graphs withntop vertices andnbottom vertices such that each top dot is connected to exactly one bottom dot and each bottom dot is connected to exactly one top dot with product given by composition σ1σ2= σ1 σ2 S3 = { , , , , , } , S2 = { , } , S1 = {} and Note that A = { , , , } is a subgroup ofS4, A = , the group generated by. Also B = , = { , , , , , , , } is a subgroup of S4.

Notes and References

These are a typed copy of Lecture 31 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 18, 2011.

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