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Math4302 Modern Algebra (Lecture 13)
Groups
Cosets
Last time we see that (left coset) a\sim b (to differentiate from right coset, we may denote it as a\sim_L b) by a^{-1}b\in H defines an equivalence relation.
Definition of Equivalence Class
Let a\in H, and the equivalence class containing a is defined as:
aH=\{x|a\simeq x\}=\{x|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}
Properties of Equivalence Class
aH=bH if and only if a\sim b.
Proof
If aH=bH, then since a\in aH, a\in bH, then for some h, a=bh, since b^{-1}a\in H, so a^{-1}b\in H, therefore a\simeq b.
If a\sim b, then aH\subseteq bH, since anything in aH is related to a, therefore it is related to b so a\in bH.
bH\subseteq aH, apply the reflexive property for equivalence relation, therefore b\in aH.
So aH=bH.
If aH\cap bH\neq \emptyset, then aH=bH.
Proof
If $x\in aH\cap bH$, then $x\sim a$ and $x\sim b$, so $a\sim b$, so $aH=bH$.aH=H if and only if a\in H.
Proof
$aH=eH$ if and only if $a\sim e$, if and only if $a\in H$.aH is called left coset of a in H.
Examples
Consider G=S_3=\{e,\rho,\rho^2,\tau_1,\tau_2,\tau_3\}.
where \rho=(123),\rho^2=(132),\tau_1=(12),\tau_2=(23),\tau_3=(13).
H=\{e,\rho,\rho^2\}.
All the left coset for H is H=eH=\rho H=\rho^2H.
\tau_1\rho=(23)=\tau_2\\
\tau_1\rho^2=(13)=\tau_3\\
\tau_2\rho=(31)=\tau_3\\
\tau_2\rho^2=(12)=\tau_1
\tau_3\rho=(12)=\tau_1\\
\tau_3\rho^2=(23)=\tau_2
\tau_1H=\{\tau_1,\tau_2,\tau_3\}=\tau_2H=\tau_3H\\
Consider G=\mathbb{Z} with H=5\mathbb{Z}.
We have 5 cosets, H,1+H,2+H,3+H,4+H.
Lemma for size of cosets
Any coset of H has the same cardinality as H.
Define \phi:H\to aH by \phi(h)=ah.
\phi is an bijection, if ah=ah'\implies h=h', it is onto by definition of aH.
Corollary: Lagrange's Theorem
If G is a finite group, and H\leq G, then |H|\big\vert |G|. (size of H divides size of G)
Proof
Suppose H has r distinct cosets, then |G|=r|H|, so |H| divides |G|.
Corollary for Lagrange's Theorem
If |G|=p, where p is a prime number, then G is cyclic.
Proof
Prick e\neq a\in G, let H=\langle a\rangle \leq G, then |H| divides |G|, since p is prime, then |H|=|G|, so G=\langle a \rangle.
If G is finite and a\in G, then \operatorname{ord}(a)\big\vert|G|.
Proof
Since \operatorname{ord}(a)=|\langle a\rangle|, and \langle a\rangle is a subgroup, so \operatorname{ord}(a)\big\vert|G|.
Definition of index
Suppose H\leq G, the number of distinct left cosets of H is called the index of H in G. Notation is (G:H).
Definition of right coset
Suppose H\leq G, define the equivalence relation by a\sim 'b (or a\sim_R b in some textbook) if a b^{-1}\in H. (note the in left coset, we use a^{-1}b \in H, or equivalently b^{-1}a \in H, these are different equivalence relations)
The equivalent class is defined
Ha=\{x\in G|x\sim'a\}=\{x\in G|xa^{-1}\in H\}=\{x|x=ha\text{ for some }h\in H\}
Some properties are the same as the left coset
Ha=H\iff a\in HHa=Hbif and only ifa\sim'b\iff a b^{-1}\in H.Ha\cap Hb\neq \emptyset\iff Ha=Hb.
Some exercises: Find all the left and right cosets of G=S_3, there should be 2 left cosets and 2 right cosets (giving different partition of G).