NoteNextra-origin/content/Math4302/Exam_reviews/Math4302_E2.md

Math 4302 Exam 2 Review

Groups

Direct products

\mathbb{Z}_m\times \mathbb{Z}_n is cyclic if and only if m and n have greatest common divisor 1.

More generally, for \mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_k}, if n_1,n_2,\cdots,n_k are pairwise coprime, then the direct product is cyclic.

If n=p_1^{m_1}\ldots p_k^{m_k}, where p_i are distinct primes, then the group

G=\mathbb{Z}_n=\mathbb{Z}_{p_1^{m_1}}\times \mathbb{Z}_{p_2^{m_2}}\times \cdots \times \mathbb{Z}_{p_k^{m_k}}

is cyclic.

Structure of finitely generated abelian groups

Theorem for finitely generated abelian groups

Every finitely generated abelian group G is isomorphic to

Z_{p_1}^{n_1}\times Z_{p_2}^{n_2}\times \cdots \times Z_{p_k}^{n_k}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}

Corollary for divisor size of abelian subgroup

If g is abelian and |G|=n, then for every divisor m of n, G has a subgroup of order m.

Warning

This is not true if G is not abelian.

Consider A_4 (alternating group for S_4) does not have a subgroup of order 6.

Cosets

Definition of Cosets

Let G be a group and H its subgroup.

Define a relation on G and a\sim b if a^{-1}b\in H.

This is an equivalence relation.

• Reflexive: a\sim a: a^{-1}a=e\in H
• Symmetric: a\sim b\Rightarrow b\sim a: a^{-1}b\in H, (a^{-1}b)^{-1}=b^{-1}a\in H
• Transitive: a\sim b and b\sim c\Rightarrow a\sim c : a^{-1}b\in H, b^{-1}c\in H, therefore their product is also in H, (a^{-1}b)(b^{-1}c)=a^{-1}c\in H

So we get a partition of G to equivalence classes.

Let a\in G, the equivalence class containing a

aH=\{x\in G| a\sim x\}=\{x\in G| a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}

This is called the coset of a in H.

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.

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)

Normal Subgroups

Definition of Normal Subgroup

A subgroup H\leq G is called a normal subgroup if aH=Ha for all a\in G. We denote it by H\trianglelefteq G

Lemma for equivalent definition of normal subgroup

The following are equivalent:

1. H\trianglelefteq G
2. aHa^{-1}=H for all a\in G
3. aHa^{-1}\subseteq H for all a\in G, that is aha^{-1}\in H for all a\in G

Factor group

Consider the operation on the set of left coset of G, denoted by S. Define

(aH)(bH)=abH

Condition for operation

The operation above is well defined if and only if H\trianglelefteq G.

Definition of factor (quotient) group

If H\trianglelefteq G, then the set of cosets with operation:

(aH)(bH)=abH

is a group denoted by G/H. This group is called the quotient group (or factor group) of G by H.

Fundamental homomorphism theorem (first isomorphism theorem)

If \phi:G\to G' is a homomorphism, then the function f:G/\ker(\phi)\to \phi(G), (\phi(G)\subseteq G') given by f(a\ker(\phi))=\phi(a), \forall a\in G, is an well-defined isomorphism.

• If G is abelian, N\leq G, then G/N is abelian.
• If G is finitely generated and N\trianglelefteq G, then G/N is finitely generated.

Definition of simple group

G is simple if G has no proper (H\neq G,\{e\}), normal subgroup.

Center of a group

Recall from previous lecture, the center of a group G is the subgroup of G that contains all elements that commute with all elements in G.

Z(G)=\{a\in G\mid \forall g\in G, ag=ga\}

this subgroup is normal and measure the "abelian" for a group.

Definition of the commutator of a group

Let G be a group and a,b\in G, the commutator [a,b] is defined as aba^{-1}b^{-1}.

[a,b]=e if and only if a and b commute.

Some additional properties:

• [a,b]^{-1}=[b,a]

Definition of commutator subgroup

Let G' be the subgroup of G generated by all commutators of G.

G'=\{[a_1,b_1][a_2,b_2]\ldots[a_n,b_n]\mid a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\in G\}

Then G' is the subgroup of G.

• Identity: [e,e]=e
• Inverse: ([a_1,b_1],\ldots,[a_n,b_n])^{-1}=[b_n,a_n],\ldots,[b_1,a_1]

Some additional properties:

• G is abelian if and only if G'=\{e\}
• G'\trianglelefteq G
• G/G' is abelian
• If N is a normal subgroup of G, and G/N is abelian, then G'\leq N.

Group acting on a set

Definition for group acting on a set

Let G be a group, X be a set, X is a $G$-set or G acts on X if there is a map

G\times X\to X

(g,x)\mapsto g\cdot x\, (\text{ or simply }g(x))

such that

1. e\cdot x=x,\forall x\in X
2. g_2\cdot(g_1\cdot x)=(g_2 g_1)\cdot x

Group action is a homomorphism

Let X be a $G$-set, g\in G, then the function

\sigma_g:X\to X,x\mapsto g\cdot x

is a bijection, and the function \phi:G\to S_X, g\mapsto \sigma_g is a group homomorphism.

Definition of orbits

We define the equivalence relation on X

x\sim y\iff y=g\cdot x\text{ for some }g

So we get a partition of X into equivalence classes: orbits

Gx\coloneqq \{g\cdot x|g\in G\}=\{y\in X|x\sim y\}

is the orbit of X.

x,y\in X either Gx=Gy or Gx\cap Gy=\emptyset.

X=\bigcup_{x\in X}Gx.

Definition of isotropy subgroup

Let X be a $G$-set, the stabilizer (or isotropy subgroup) corresponding to x\in X is

G_x=\{g\in G|g\cdot x=x\}

G_x is a subgroup of G. G_x\leq G.

• e\cdot x=x, so e\in G_x
• If g_1,g_2\in G_x, then (g_1g_2)\cdot x=g_1\cdot(g_2\cdot x)=g_1 \cdot x, so g_1g_2\in G_x
• If g\in G_x, then g^{-1}\cdot g=x=g^{-1}\cdot x, so g^{-1}\in G_x

Orbit-stabilizer theorem

If X is a $G$-set and x\in X, then

|Gx|=(G:G_x)=\text{ number of left cosets of }G_x=\frac{|G|}{|G_x|}

Theorem for orbit with prime power groups

Suppose X is a $G$-set, and |G|=p^n for some prime p. Let X_G be the set of all elements in X whose orbit has size 1. (Recall the orbit divides X into disjoint partitions.) Then |X|\equiv |X_G|\mod p.

Corollary: Cauchy's theorem

If p is prime and p|(|G|), then G has a subgroup of order p.

This does not hold when p is not prime.

Consider A_4 with order 12, and A_4 has no subgroup of order 6.

Corollary: Center of prime power group is non-trivial

If |G|=p^m, then Z(G) is non-trivial. (Z(G)\neq \{e\})

Proposition: Prime square group is abelian

If |G|=p^2, where p is a prime, then G is abelian.

Classification of small order

Let G be a group

• |G|=1
  • G=\{e\}
• |G|=2
  • G\simeq\mathbb{Z}_2 (prime order)
• |G|=3
  • G\simeq\mathbb{Z}_3 (prime order)
• |G|=4
  • G\simeq\mathbb{Z}_2\times \mathbb{Z}_2
  • G\simeq\mathbb{Z}_4
• |G|=5
  • G\simeq\mathbb{Z}_5 (prime order)
• |G|=6
  • G\simeq S_3
  • G\simeq\mathbb{Z}_3\times \mathbb{Z}_2\simeq \mathbb{Z}_6

Proof

|G| has an element of order 2, namely b, and an element of order 3, namely a.

So e,a,a^2,b,ba,ba^2 are distinct.

Therefore, there are only two possibilities for value of ab. (a,a^2 are inverse of each other, b is inverse of itself.)

If ab=ba, then G is abelian, then G\simeq \mathbb{Z}_2\times \mathbb{Z}_3.

If ab=ba^2, then G\simeq S_3.

• |G|=7
  • G\simeq\mathbb{Z}_7 (prime order)
• |G|=8
  • G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2
  • G\simeq\mathbb{Z}_4\times \mathbb{Z}_2
  • G\simeq\mathbb{Z}_8
  • G\simeq D_4
  • G\simeq quaternion group \{e,i,j,k,-1,-i,-j,-k\} where i^2=j^2=k^2=-1, (-1)^2=1. ij=l, jk=i, ki=j, ji=-k, kj=-i, ik=-j.
• |G|=9
  • G\simeq\mathbb{Z}_3\times \mathbb{Z}_3
  • G\simeq\mathbb{Z}_9 (apply the corollary, 9=3^2, these are all the possible cases)
• |G|=10
  • G\simeq\mathbb{Z}_5\times \mathbb{Z}_2\simeq \mathbb{Z}_{10}
  • G\simeq D_5
• |G|=11
  • G\simeq\mathbb{Z}_11 (prime order)
• |G|=12
  • G\simeq\mathbb{Z}_3\times \mathbb{Z}_4
  • G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3
  • A_4
  • D_6\simeq S_3\times \mathbb{Z}_2
  • ??? One more
• |G|=13
  • G\simeq\mathbb{Z}_{13} (prime order)
• |G|=14
  • G\simeq\mathbb{Z}_2\times \mathbb{Z}_7
  • G\simeq D_7

Lemma for group of order 2p where p is prime

If p is prime, p\neq 2, and |G|=2p, then G is either abelian \simeq \mathbb{Z}_2\times \mathbb{Z}_p or G\simeq D_p

Ring

Definition of ring

A ring is a set R with binary operation + and \cdot such that:

• (R,+) is an abelian group.
• Multiplication is associative: (a\cdot b)\cdot c=a\cdot (b\cdot c).
• Distribution property: a\cdot (b+c)=a\cdot b+a\cdot c, (b+c)\cdot a=b\cdot a+c\cdot a. (Note that \cdot may not be abelian, may not even be a group, therefore we need to distribute on both sides.)

Note

a\cdot b=ab will be used for the rest of the sections.

Properties of rings

Let 0 denote the identity of addition of R. -a denote the additive inverse of a.

• 0\cdot a=a\cdot 0=0
• (-a)b=a(-b)=-(ab), \forall a,b\in R
• (-a)(-b)=ab, \forall a,b\in R

Definition of commutative ring

A ring (R,+,\cdot) is commutative if a\cdot b=b\cdot a, \forall a,b\in R.

Definition of unity element

A ring R has unity element if there is an element 1\in R such that a\cdot 1=1\cdot a=a, \forall a\in R.

Definition of unit

Suppose R is a ring with unity element. An element a\in R is called a unit if there is b\in R such that a\cdot b=b\cdot a=1.

In this case b is called the inverse of a.

Definition of division ring

If every a\neq 0 in R has a multiplicative inverse (is a unit), then R is called a division ring.

Definition of field

A commutative division ring is called a field.

Units in \mathbb{Z}_n is coprime to n

More generally, [m]\in \mathbb{Z}_n is a unit if and only if \operatorname{gcd}(m,n)=1.

Integral Domains

Definition of zero divisors

If a,b\in R with a,b\neq 0 and ab=0, then a,b are called zero divisors.

Zero divisors in \mathbb{Z}_n

[m]\in \mathbb{Z}_n is a zero divisor if and only if \operatorname{gcd}(m,n)>1 (m is not a unit).

Corollaries of integral domain

If R is a integral domain, then we have cancellation property ab=ac,a\neq 0\implies b=c.

Units with multiplication forms a group

If R is a ring with unity, then the units in R forms a group under multiplication.

Fermat’s and Euler’s Theorems

Fermat’s little theorem

If p is not a divisor of m, then m^{p-1}\equiv 1\mod p.

Corollary of Fermat’s little theorem

If m\in \mathbb{Z}, then m^p\equiv m\mod p.

Euler’s totient function

Consider \mathbb{Z}_6, by definition for the group of units, \mathbb{Z}_6^*=\{1,5\}.

\phi(n)=|\mathbb{Z}_n^*|=|\{1\leq x\leq n:gcd(x,n)=1\}|

Euler’s Theorem

If m\in \mathbb{Z}, and gcd(m,n)=1, then m^{\phi(n)}\equiv 1\mod n.

Theorem for existence of solution of modular equations

ax\equiv b\mod n has a solution if and only if d=\operatorname{gcd}(a,n)|b And if there is a solution, then there are exactly d solutions in \mathbb{Z}_n.

Ring homomorphisms

Definition of ring homomorphism

Let R,S be two rings, f:R\to S is a ring homomorphism if \forall a,b\in R,

• f(a+b)=f(a)+f(b)\implies f(0)=0, f(-a)=-f(a)
• f(ab)=f(a)f(b)

Definition of ring isomorphism

If f is a ring homomorphism and a bijection, then f is called a ring isomorphism.