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Subgroups and Quotient Groups

Basic Definitions

Generating sets

Definition: Suppose G is a group and A is a subset of G (not necessarily a subgroup, just a bunch of elements). The subgroup A of G generated by A is A=H where the intersection is over all subgroups H of G that contain the set A.

Some special types of subgroups

See DF Section 2.2

Suppose G is a group and H is a subgroup of G.

  1. The centralizer CG(H) of H is the set of elements gG such that gh=hg for all hH.
  2. The normalizer NG(H) of H is the set of elements gG such that gHg1=H. In other words, ghg1H for all hH.
  3. H is a normal subgroup if NG(H)=G.
  4. The center Z(G) of G is the the of elements z of G such that zg=gz for all gG.
  5. If f:GH is a homomorphism, the kernel of f is the set of gG such that f(g)=e.

Notice that:

  1. CG(H)NG(H)
  2. The center Z(G) is a normal subgroup of G.
  3. HNG(H) and H is a normal subgroup of NG(H).
  4. The kernel of any homomorphism is a normal subgroup. (In fact, the converse is true as well, as we will see later).

Subgroups from group actions

Suppose that X is a set and G acts on X. Remember that one way to think of this is that we have a homomorphism from G to S(X). Another way is that we have a map G×XX satisfying ex=x and g(h(x))=(gh)(x) for all xX and all g,hG. Such an action yields subgroups of G as follows:

  1. The kernel of the action is the set of gG such that gx=x for all xX. In other words, the kernel of the action is the kernel of the homomorphism from G to S(X) corresponding to the action. The kernel of the action is therefore a normal subgroup of G.
  2. If xX, the set of elements gG such that gx=x is a subgroup of G called the stabilizer of x.

Normalizers and centralizers via group actions

One way to think of the normalizer of H in G as being the largest subgroup of G in which H is normal. Alternatively one can think of them in terms of group actions.

Let P(G) be the power set of G – that is, the set of subsets of G. If SP(G) is a subset, define g(S)={gsg1:sS}. This defines an action of G on P(S). In general the operation hghg1 is called conjugation of h by g.

If we choose S=H, then by definition NG(H) is exactly the stabilizer of H for this action.

If we restrict the action of NG(H) to the set H, then CG(H) is exactly the subset of NG(H) that fixes H pointwise. In other words, CG(H) is the kernel of the conjugation action of NG(H) on H.

Cosets

See DF Section 3.1

Definition: Let H be a subgroup of a group G. A set gH={gh:hH} is called a left coset of H. The corresponding set Hg is called a right coset.

Basics

  1. gH=H if and only if gH.
  2. Any two left (right) cosets are in bijection with each other, so if H is finite all cosets of H have the same number of elements as H.
  3. Two left (right) cosets are either equal or disjoint.
  4. There is a bijection between the set of left cosets and the set of right cosets of H in G given by f(gH)=Hg1.
  5. Together, the left (right) cosets of G form a partition of G into disjoint sets.
  6. The index of H in G, written [G:H], is the number of left (right) cosets, if that number is finite; otherwise we say H has infinite index in G.
  7. H is normal if and only if gH=Hg for every gG.

One way to obtain most ofthe key properties of cosets is to observe that the relation xy defined by x=yh for some hH – or, expressed another way, that xyH – is an equivalence relation.

Theorem: (Lagrange) If G is finite and H is a subgroup of G then H[G:H]=∣G.

Corollary: In a finite group, the order of an element divdes the order of the group.

Quotient Group

If H is a normal subgroup, then the set of left (right) cosets of G form a group called the quotient group G/H. The group law is (aH)(bH)=(ab)H.

The key ingredient of this definition is that the product is well defined. In other words, if aH=aH and bH=bH then (ab)H=(ab)H. Since H is normal, xH=Hx for any xG. We have a=ah and b=bk for elements h,k in H. Then ahbk=abhk since Hb=bH by normality of H. This shows that abH=abH.

Universal property

Let πH:GG/H be the “canonical map” that sends ggH.

  • The kernel of this map is H.
  • Let K be any group and let f:GK be a homomorphism such that H is contained in the kernel of f. Then there is a unique homomorphism f:G/HK such that f=fπG.

The map f is defined by f(aH)=f(a). This is well defined since f(h)=e for all hH.

Corollary: A subgroup H is normal if and only if it is the kernel of a homomorphism.

In fact if H is normal then H is the kernel of the homomorphism πG.