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Cosets

Let GG be a group, and let H⊂GH\subset G be a subset(subgroup better). ∀a∈G\forall a\in G:

  • aH={ah:h∈H}aH=\{ah:h\in H\} is the left coset of HH containing aa
  • Ha={ha:h∈H}Ha=\{ha:h\in H\} is the right coset of HH containing aa
  • aHa−1={aha−1:h∈H}aHa^{-1}=\{aha^{-1}:h\in H\} (the the conjugate of HH with respect to aa)
  • such aa above is called a coset representative

Lemma: Now consider H⊂G,a,b∈GH\subset G, a,b\in G, then we have following properties:

  1. a∈aHa\in aH
  2. aH=H  ⟺  a∈HaH = H \iff a\in H
  3. (ab)H=a(bH)(ab)H = a(bH) and H(ab)=(Ha)bH(ab) = (Ha)b
  4. aH=bH  ⟺  a∈bH  ⟺  a−1b∈HaH=bH \iff a\in bH \iff a^{-1}b\in H
    • similarly, Ha=Hb  ⟺  a∈Hb  ⟺  ba−1∈HHa = Hb \iff a\in Hb \iff ba^{-1}\in H
  5. (aH=bHaH = bH or aH∩bH=∅aH\cap bH = \empty) and (Ha=HbHa = Hb or Ha∩Hb=∅Ha\cap Hb = \empty)
  6. ∣aH∣=∣bH∣|aH| = |bH|
  7. aH=Ha  ⟺  H=aHa−1=a−1HaaH = Ha \iff H = a H a^{-1} = a^{-1}Ha
  8. aHaH is a subgroup of HH   ⟺  a∈H\iff a\in H
  9. (aH)−1=Ha−1(aH)^{-1} = Ha^{-1}
  10. HH is fixed by all inner automorphisms of G

Lagrange's Theorem​

Lagrange's theorem:

  • ∣G∣<∞,H≤G  ⟹  ∣H∣∣∣G∣|G| < \infty, H \le G \implies |H| | |G|
  • the number of left/right cosets of HH in GG is ∣G∣/∣H∣|G|/|H|

Then we have following corollaries, before that we need have some definitions. WE define the number of distinct left cosets of HH in GG as ∣G:H∣|G: H| and called it index.

  • The second point from lagrange's theorem can be wrote as ∣G∣<∞,H≤G  ⟹  ∣G:H∣=∣G∣/∣H∣|G| < \infty, H \le G \implies |G: H| = |G|/|H| now.
  • ∣G∣<∞,∀a∈G,∣a∣∣∣G∣|G| < \infty, \forall a\in G, |a| | |G|
  • A group of prime order is cyclic.
  • ∣G∣<∞,a∈G  ⟹  a∣G∣=e|G| < \infty, a\in G \implies a^{|G|} = e
  • Fermat's Little Theorem: ap≡a(mod p)a^p \equiv a(mod \ p), p is a prime

An Application of Cosets to Permutation Groups​

Let GG be a permutation group of a set SS. For each i∈Si\in S,

  • we define stabG(i)={ϕ∈G:ϕ(i)=i}stab_G(i) = \{\phi\in G : \phi(i) = i\}. We call stabG(i)stab_G(i) the stabilizer of ii under GG.
  • we define orbG(i)={ϕ(i):ϕ∈G}orb_G(i) = \{\phi(i): \phi\in G\}. We call orbG(i)orb_G(i) the orbit of ii under GG.

Theorem​

Let H,KH, K be two finite subgroups of a group GG. Define HK={hk:h∈H,k∈K}  ⟹  ∣HK∣=∣H∣∣K∣∣H∩K∣HK = \{hk:h\in H, k\in K\} \implies |HK| =\frac{|H||K|}{|H\cap K|}

  • HK=KH  ⟺  HKHK = KH \iff HK is a subgroup in GG

Theorem​

Every group of order 2p2p for a prime p>2p > 2 is isomorphic to Z2p\Z_{2p} or DpD_p

Theorem​

∣G∣=2p  ⟹  G|G| = 2p \implies G contains an element of order pp.