# MAT247H1 Chapter Notes - Chapter 2: Non-Abelian Group, Invariant Subspace, Binary Operation

by OC118869

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MathematicsCourse Code

MAT247H1Professor

Fiona T RahmanChapter

2This

**preview**shows page 1. to view the full**5 pages of the document.**Mat 247 - Deﬁnitions and results on group theory

Deﬁnition: Let Gbe nonempty set together with a binary operation (usually called mul-

tiplication) that assigns to each pair of elements g1,g2∈Gan element in G, denoted

by g1g2or g1·g2. We say that Gis a group under this operation if the following three

properties are satisﬁed:

Associativity: (g1g2)g3=g1(g2g3) for all g1,g2, and g3∈G.

Existence of identity element: There exists an element e(called an identity) in Gsuch

that g·e=e·g=gfor all g∈G.

Existence of inverses: Let ebe an identity element in G. For each element g∈G,

there is an element g−1∈G(called an inverse of g) such that g·g−1=g−1·g=e.

Examples: (details omitted)

(1) If Fis a ﬁeld and nis a postive integer, let GLn(F) = {A∈Mn×n(F)|det(A)6= 0 }.

Then GLn(F) is a group under the operation of matrix multiplication.

(2) Let Vbe a ﬁnite-dimensional vector space over a ﬁeld F. Let G={T∈ L(V)|Tis invertible }.

Then GL(V) is a group under the operation of composition of linear transformations.

(3) Let nbe a positive integer. The set Unof (complex) unitary n×nmatrices is a group

under the operation of matrix multiplication.

(4) The set Zof integers is a group under the operation of addition of integers. (Note:

e= 0; the inverse of m∈Zis −m.)

(5) The set Z\{0}of nonzero integers is not a group under the operation of multiplication

of integers. The operation is associative and 1 is an identity, but the only nonzero

integers that have inverses in Z\{0}are 1 and −1.

Deﬁnition: If Gis a group, we say that Gis abelian (or commutative) if g1g2=g2g1for

all g1and g2∈G. If Gis not abelian, we say that Gis nonabelian (or noncommutative).

Deﬁnition: The order of a group Gis the number of elements in G. If the order of Gis

ﬁnite, we say that Gis a ﬁnite group. Otherwise, we say that Gis an inﬁnite group.

If Gis an abelian group, the group operation may be written with a plus sign: g1+g2

instead of g1g2.

Examples. If Fis a ﬁnite ﬁeld, then GLn(F) is a ﬁnite group. If Fis an inﬁnite ﬁeld, then

GLn(F) is an inﬁnite group. If n≥2, then GLn(F) is a nonabelian group. The notation

F×is often used for the group GL1(F) of nonzero elements in F(with the operation of

multiplication in F). The group F×is abelian.

Lemma. If Gis a group, there is a unique identity element in G. If g∈G, there is a

unique inverse g−1of gin G.

Proof. If eand e0are identity elements in G, we have e·e0=e0·e=e, using that e0is

an identity element, and we also have e·e0=e0·e=e0, since eis an identity element.

Therefore e·e0=e=e0. The second part is left as an exercise.

Deﬁnition. If His a (nonempty) subset of a group Gand His itself a group under the

operation on G, we say that His a subgroup of G.

The subset {e}of a group Gis a subgroup of G. Clearly, Gis a subgroup of G. The

proof of the following lemma was discussed in class.

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Lemma. (Subgroup test) A nonempty subset Hof a group Gis a subgroup of Gif and

only if h1h−1

2∈Hfor all h1,h2∈H.

Examples:

(1) Let G=GLn(F), n≥2, and let H={A∈G|Ajk = 0 whenever j > k }. Then H

is a subgroup of G. (Details omitted.)

(2) Let Vbe a vector space of dimension n≥2, let G=GL(V), and let H={T∈

G|nullity(T−IV)>0}. Let β={x1, . . . , xn}be an ordered basis for V. There exists

a unique T∈ L(V) such that T(x1) = x1and T(xj) = −xj, 2 ≤j≤n. Check that T

is invertible, nullity(T−IV) = 1, −Tis invertible, and nullity(−T−IV) = n−1 (left

as an exercise). This implies that T,−T∈H. But (T◦(−T))(xj) = −T2(xj) = xj

for 1 ≤j≤n, so T◦(−T) = −IV, and nullity(−IV−IV) = nullity(−2·IV) = 0.

That is, T◦(−T)/∈H. This implies that His not a subgroup of G.

(3) Let G=GLn(F), n≥2. Let Dnbe the set of diagonal matrices in G. Then Dnis a

subgroup of G, and Dnis abelian. (This example shows that there can be nontrivial

abelian subgroups of nonabelian groups.)

Deﬁntion. A subgroup Hof a group Gis said to be normal in Gif ghg−1∈Hfor all

g∈Gand h∈H.

Examples: (details omitted)

(1) Let G=GL2(F) and let H={A∈G|A21 = 0 }. Then His a subgroup of Gbut

His not normal in G. (Note that h=1 1

0 1 ∈H, Let g=0 1

1 0 . Show that

g∈Gand ghg−1/∈H.)

(2) Let G=GLn(F),n≥2, and let H=SLn(F) = {A∈G|det(A) = 1 }. Then His a

normal subgroup of G. (This is easily proved using properties of determinants.)

(3) If Gis an abelian group, then any subgroup Hof Gis normal in Gbecause ghg−1=

h(g·g−1) = h·e=hfor all h∈Hand g∈G.

Deﬁnition. If Gand G0are groups, a map ϕ:G→G0is a homomorphism if ϕ(g1g2) =

ϕ(g1)ϕ(g2) for all g1and g2∈G.

Examples. (details omitted)

(1) Then det : GLn(F)→F×=GL1(F) is a homomorphism.

(2) If Gis a nonabelian group, the map ϕ:G→Gdeﬁned by ϕ(g) = g2is not a

homomorphism. (Here, g2=g·g,g∈G.)

Notation. If Gis a group, g∈G, and n∈Z, deﬁne g0=e,gn=g·gn−1,n≥1, and

gn= (g−1)−n,n≤ −1.

Lemma. Let Gand G0be groups and let ϕ:G→G0be a homomorphism.

(1) Let eand e0be identity elements in Gand G0, respectively. Then ϕ(e) = e0.

(2) If g∈Gand n∈Z, then ϕ(gn) = (ϕ(g))n.

Deﬁnition: Let Gand G0be groups and let ϕ:G→G0be a homomorphism.

(1) The kernel of ϕis deﬁned to be {g∈G|ϕ(g) = e0}. Here, e0is the identity element

in G0.

(2) The image of ϕis deﬁned to be ϕ(G) = {ϕ(g)|g∈G}.

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