Math 541 - 2/19

Regular n\gon • A regular n\gon is a polygon with all sides and angles equal Symmetry • Definition ○ A symmetry of a regular n-gon is a way of § picking up a copy of it § moving it around in 3d § setting it back down ○ so that it exactly covers the original • Examples ○ Rotations ○ Reflection Dihedral Groups (Section 1.2) • Definition ○ D_2n≔{symmetries of the n-gon} is called n-th dihedral groups • Note ○ |D_2n |=2n (proof on page 24) ○ There are n rotations and n reflections ○ Symmetries of n-gons are determined by ○ the permutations of the vertices they induce • Example: n=3 ○ Rotations § 120°:(1 2 3) § 240°:(1 3 2) § 360°:(1) ○ Reflections § (2 3) § (1 3) § (1 2) ○ D_6≅{(1),(2 3),(1 3),(1 2),(1 3 2),(1 2 3)}=S_3 • Example: n=4 ○ Rotations § 90°:(1 2 3 4) § 180°:(1 3)(2 4) § 270°:(1 4 3 2) § 360°:(1) ○ Reflections § (2 4) § (1 3) § (1 4)(2 3) § (1 2)(3 4) ○ D_8≅{(1),(1 2 3 4),(1 3)(2 4),(1 4 3 2),(1 3),(2 4),(1 4)(2 3),(1 2)(3 4)}≤S_4 • Fact ○ In general D_2n is isomorphic to a subgroup of S_n ○ Every finite group is isomorphic to a subgroup of a symmetric group Proposition 17 (The Subgroup Criterion) • Statement ○ A subset H of a group G is a subgroup iff ○ H≠∅ ○ ∀x,y∈H, xy^(−1)∈H • Recall the original definition ○ A subset H of a group G is a subgroup iff ○ H≠∅ ○ ∀h,h′∈H, hh′∈H ○ ∀h∈H, h(−1)∈H • Proof ○ (⟹) Clear ○ (⟸) We must check that H is closed under multiplication and inversion ○ Let x∈H ○ 1⋅x^(−1)∈H ○ Thus x^(−1)∈H ○ Let y∈H, then y^(−1)∈H ○ So x(y^(−1) )^(−1)∈H ○ xy∈H Examples of Subgroups • Example 1 ○ Z≤Q≤R≤ℂ • Example 2 ○ Definition § Fix n∈Z( 0) § SL_n (R≔{A∈GL_n (R│det⁡A=1} is called the special linear group ○ Claim § SL_n (R≤GL_n (R ○ Proof § SL_n (R≠∅, since I_n∈SL_n (R § Let A,B∈SL_n (R § det⁡(AB^(−1) )=det⁡A⋅det⁡〖B^(−1) 〗=det⁡A/det⁡B =1/1=1∎ • Example 3 ○ Definition § If G is a group § Z(G)≔{a∈G│ag=ga, ∀g∈G} is called the center or G ○ Claim § Z(G)≤G ○ Proof § Z(G)≠∅, since 1∈Z(G) § Let a,b∈Z(G) § If g∈G, abg=agb=gab § so Z(G) is closed under multiplication § Also a^(−1) g=(g^(−1) a)^(−1)=(ag(−1) )^(−1)=ga(−1) § so Z(G) is closed under inversion Cyclic Groups • Definition ○ A group G is cyclic if ∃g∈G s.t. ⟨g⟩=G ○ In this case, we say G is generated by g • Note ○ If G is finite of order n, then ○ G is cyclic iff ∃g∈G s.t. |g|=n