Note of Michael Artin "Algebra" Chapter 2 "Groups"

2.1 LAWS OF COMPOSITION

Def. (Laws of Composition)

Any rule combining pairs of element of \(S\) to get another element of \(S\): \(S\times S \to S\).

2.1.1 Def. (Associative Law) (Skipped)

2.1.2 Def. (Commutative Law) (Skipped)

2.1.4 Prop. (Associative Law \(\Rightarrow\) Product Def.)

\([a_1] = a_1,[a_1a_2] = a_1\times a_2,\forall \text{ integer } i \in [1,n],[a_1\cdots a_n] = [a_1\cdots a_i][a_{i+1}\cdots a_n]\)

2.1.5 Def. (Identity)

\(ea=a\) and \(ae=a,\forall a \in S\)

• Left identity and right identity: Something deters whether a semi-group can be a group.(to do)

2.2 GROUPS AND SUBGROUPS

Def. (Group)

• Associativity, identity, inverse(both left and right)
• Abelian group (Group + Commutativity)

2.2.1 Def. (Order of Group)

\(|G|\) = number of elements of \(G\).

2.2.2 Def. (Familiar Infinite Abelian Groups)

\(\mathbb{Z}^+, \mathbb{R}^+, \mathbb{R}^{\times}, \mathbb{C}^+, \mathbb{C}^{\times}\)

2.2.3 Prop. (Cancellation Law)

If \(ab=ac\) or if \(ba=ca\), the \(b=c\). If \(ab=a\) or if \(ba=a\), then \(b=1\).

2.2.4 Def. (General Linear Group)

\(GL_n = \left\{ A_{n\times n} | det(A) \neq 0 \right\}\)
Working on \(\mathbb{R}\) or \(\mathbb{C}\), we have \(GL_n(\mathbb{R}), GL_n(\mathbb{C})\).

2.2.5 Def. (Symmetric Group \(S_n\))

\(S_n\) is the group of permutations of the set of indices \({1,2,\cdots, n}\).

2.2.9 Def. (Subgroup)

\(G\) is a group, \(H \subset G\) is a subgroup if it has the following properties:

• Closure: If \(a\) and \(b\) are in \(H\), then \(ab\) is in \(H\).
• Identity: \(1\) is in \(H\).
• Inverses: If \(a\) is in \(H\), then \(a^{-1}\) is in H.

2.2.10 Example. (Circle Group, Special Linear Group, Trivial Subgroup and Proper Subgroup)

Circle group

Circle group is \(\left\{ x| x\in \mathbb{C}, |x| = 1\right\}\), the set of points on the unit circle in the complex plane, is a subgroup of \(\mathbb{C}^{\times}\).

Special Linear Group \(SL_n\)

\(SL_n(\mathbb{R})\) is the set of real \(n\times n\) matrices \(A\) with determinant equal to \(1\).

Trivial Subgroup and Proper Subgroup

• Trivial Subgroup: \(\left\{ 1 \right\}, G \leq G\)
• Proper Subgroup: Subgroups not trivial.

2.3 SUBGROUPS OF THE ADDITIVE GROUP OF INTEGERS

2.3.1 Def. (Subgroup by additive notation) (Skipped)

2.3.2 Notation. (\(\mathbb{Z}a\))

\(\mathbb{Z}a\) is a subset of \(\mathbb{Z}\) that consists of all multiples of \(a\) by \(\mathbb{Z}a\).

2.3.3 Theorem.

Let \(S \leq \mathbb{Z}^+\), then \(S\) is trivial or \(S = \mathbb{Z}a\), where \(a\) is the smallest positive integer in \(S\).

2.3.4, 2.3.5 Prop.

\[S = \mathbb{Z}a + \mathbb{Z}b = \left\{n\in\mathbb{Z} | n=ra+sb \text{ for some integers } r,s \right\} = \mathbb{Z}d, d = \gcd(a,b) \]

2.3.6 Prop. 2.3.8

\[\mathbb{Z}a\cap \mathbb{Z}b = \mathbb{Z}m \Rightarrow m = \text{lcm} (a,b) \]

2.4 CYCLIC GROUPS

2.4.1 Def. (Cyclic Group)

\(x\) is an arbitrary element of \(G\), the cyclic group generated by \(x\) is \(H = \left\{ \cdots, x^{-2}, x^{-1}, 1, x, x^2, \cdots\right\}.\) We have \(H \leq G\), and \(H\) is the smallest subgroup of \(G\) that contains \(x\).

2.4.2 Prop.

Let <\(x\)> be the cyclic subgroup of \(G\) generated by \(x \in G\), and let \(S\) denote the set of integers \(k\) such that \(x^k = 1\).

• (a) The set \(S\) is a subgroup of the additive group \(\mathbb{Z}\).
• (b) Two powers \(x^r=x^s\), with \(r \geq s\), are equal if and only if \(x^{r-s} = 1\), i. e., if and only if \(r-s\) is in \(S\).
• (c) Suppose that \(S\) is proper subgroup. Then \(S = \mathbb{Z}n\) for some positive integer \(n\). The powers \(1,x,x^2,\cdots,x^{n-1}\) are the distinct elements of the subgroup <\(x\)>, and the order of <\(x\)> is \(n\).

The group <\(x\)> described by part (c) is called the cyclic group of order \(n\). An element \(x\) of a group has order \(n\) if \(n\) is the smallest positive integer with the property \(x^n=1\), which is the same thing as saying that the cyclic subgroup <\(x\)> generated by \(x\) has order \(n\).

2.4.4 Example. (Klein four group \(V\))

\[V = \left[ \begin{matrix} \pm 1 & \\ & \pm 1 \end{matrix} \right] \]

\(V\) is the simplest group that is not cyclic.

2.4.5 Example. (Quaternion group \(H\))

\[H = \left\{ \pm 1, \pm i, \pm j, \pm k\right\} \]

where

\[\textbf{1} = \left[ \begin{matrix} 1 & 0\\ 0 & 1 \end{matrix} \right], \textbf{i} = \left[ \begin{matrix} i & 0\\ 0 & -i \end{matrix} \right], \textbf{j} = \left[ \begin{matrix} 0 & 1\\ -1 & 0 \end{matrix} \right], \textbf{k} = \left[ \begin{matrix} 0 & i\\ i & 0 \end{matrix} \right] \]

These matrices can be obtained from the Pauli matrices of physics by multiplying by \(i\).

The two elements \(i,j\) generate \(H\). \(i^2=j^2=k^2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j\).

2.5 HOMOMORPHISMS

2.5.1, 2.5.3 Def. (Homomorphism)

A homomorphism \(\varphi : G \to G'\) is a map such that \(\forall a, b \in G\), \(\varphi (ab) = \varphi (a) \varphi (b)\).

Group homomorphism maps product of elements, identity and inverses to themselves.

2.5.4, 2.5.5 Def. (Image and Kernel) (Skipped)

2.5.6 Example.

• Kernel of determinant homomorphism: \(\textbf{ker}( GL_n ( \mathbb{R}) \to \mathbb{R}^{\times}) = SL_n(\mathbb{R})\)
• Kernel of sign homomorphism: \(\textbf{ker}(S_n\to \left\{ \pm 1 \right\}) = A_n\)
  • The alternating group \(A_n\) is the group of even permutations.

2.5.7 Def. (Coset) (Skipped)

2.5.8 Prop.

\(\varphi:G \to G'\) is homomorphism of groups, \(a,b\in G, K = \textbf{ker} \varphi\). Following conditions are equivalent:

• \(\varphi(a) = \varphi(b)\)
• \(a^{-1}b \in K\)
• \(b \in aK\)
• \(aK = bK\)

2.5.10 Def. (Normal Subgroup)

A subgroup \(N \leq G\) is a normal subgroup if \(\forall a \in N, \forall g \in G, \text{ the conjugate } gag^{-1}\in N\).

2.5.11 Prop.

The kernel of a homomorphism is a normal subgroup.

2.5.12 Def. (Group Center)

A group center is the set of element that commute with every element in \(G\).

\[Z = \left\{ z\in G | zx=xz, \forall x \in G\right \}, Z \trianglelefteq G \]

2.6 ISOMORPHISMS

2.6.1 Def. (Isomorphism)

Bijective group homomorphism.

Notation: \(G \approx G'\)

2.6.2 Lemma.

\(\varphi: G \to G'\) is isomorphism \(\leftrightarrow \varphi^{-1}: G' \to G\) is isomorphism.

2.6.4 Def. (Automorphism)

\(\varphi: G \to G\) is isomorphism. The most important automorphism: Conjugation by \(g\) : \(\varphi(x) = gxg^{-1}, g, x \in G\)

2.7 EQUIVALENCE RELATIONS AND PARTITIONS

2.7.3 Def. (Equivalence)

\(a \sim b, a,b \in S\) requires transitive, symmetric, reflexive.

2.7.4 Prop.

A equivalence relation on a set \(S\) determines a partition of \(S\), and conversely.
Definition of equivalence class of \(a\):

\[C_a = \left\{ b\in S | a \sim b\right \} \]

2.7.6 Lemma.

Given an equivalence relation on a set \(S\), the subsets of \(S\) that are equivalence classes partition \(S\).
If \(C_a\) and \(C_b\) have an element in common, then \(C_a = C_b\).
Set \(\bar{S}\) contains \(S\)'s subset elements. If \(U\subseteq S\), we denote that \([U] \in \bar{S}\).

2.7.11 Def. (Inverse image)

Any map of sets \(f:S\to T\) gives us an equivalence relation on \(S\): \(a\sim b\) if \(f(a)=f(b)\).

Definition of inverse image of an element \(t\) of \(T\):

\[f^{-1} (t) = \left\{ s \in S | f(s)=t \right\} \]

\(f\) is bijective if and only if \(f\) is a map. The inverse images are also called the fibres of the map \(f\), and the fibres that are not empty are the equivalence classes for the relation defined above.

2.7.15 Prop.

Let \(K\) be the kernel of a homomorphism \(\varphi:G \to G'\). The fibre of \(\varphi\) that contains an element \(a\) of \(G\) is the coset \(aK\) of \(K\). These cosets partition the group \(G\), and they correspond to elements of the image of \(\varphi\).

2.8 COSETS

2.8.1 Def. (Coset)

\[H \leq G, a \in G, aH = \left\{ ah| h \in H \right \} \]

The cosets of \(H \leq G\) are equivalent classes for the congruence relation \(a \equiv b\) if \(b=ah, h\in H\).

2.8.3 Corollary.

The left cosets of a subgroup \(H \leq G\) partition the group.

The following are equivalent:

• \(b=ah\) for some \(h\in H\), or, \(a^{-1}b \in H\).
• \(b \in aH\)
• \(aH=bH\)

2.8.6 Def. (Index of Left Cosets)

The number of left cosets of a subgroup is called the index of \(H\) in \(G\), denoted by \([G:H]\).

2.8.7 Lemma.

All left cosets \(aH\) of a subgroup \(H\) of a group \(G\) have the same order.

2.8.8 Corollary. (Counting Formula)

\[|G| = |H| [G:H] \]

2.8.9 Theorem. (Lagrange's Theorem)

Let H be a subgroup of a finite group G. The order of H divides the order of G.

2.8.10 Corollary.

The order of an element of a finite group divides the order of the group. <\(a\)> \(\leq G, a\in G\) (Prop. 2.4.2)

2.8.11 Corollary.

Suppose that a group \(G\) has prime order \(p\). Let \(a\) be any element of \(G\) other than the identity. Then \(G\) is the cyclic group <\(a\)> generated by \(a\).

2.8.12 Corollary.

Referring to 2.7.15, we can conclude a bijective correspondence with the elements of the image.

\[[G: \ker \varphi] = |\text{im} \varphi| \]

2.8.13 Corollary.

Let \(\varphi: G \to G'\) be a homomorphism of finite groups. Then

• \(|G| = |\ker\varphi | \cdot |\text{im} \varphi|\)
• \(|\ker\varphi |\) divides \(|G|\), and
• \(|\text{im}\varphi|\) divides both \(|G|\) and \(|G'|\).

2.8.14 Prop. (Multiplicative Property of the Index)

Let \(G\supset H \supset K\) be subgroups of a group \(G\). Then \([G:K]=[G:H][H:K]\).

2.8.15 Def. (Right Cosets)

The partitions of right cosets of a subgroup of a group are not the same as those of the left cosets. However, if a subgroup is normal, the right coset is equal to the left coset.

2.8.17 Prop.

Let \(H\) be a subgroup of a group \(G\). The following conditions are equivalent:

• (i) \(H\) is a normal subgroup: \(\forall h \in H\) and all \(g\in G\), \(ghg^{-1}\) is in \(H\).
• (ii) \(\forall g\in G, gHg^{-1} = H\).
• (iii) \(\forall g \in G, gH = Hg\).
• (iv) Every left coset \(H\leq G\) is a right coset.
  The circumstance of the left coset be equal to the right coset: Let \(H\leq G\), \(g \in gH \cap Hg\), and the right coset partition the group \(G\), therefore if the left coset \(gH\) is equal to any right coset, that coset must be \(Hg\).

2.8.18 Prop.

• (a) If \(H\) is a subgroup of a group and \(g\) is an element of \(G\), the set \(gHg^{-1}\) is also a subgroup.
• (b) If a group \(G\) has just one subgroup \(H\) of order \(r\), then that subgroup is normal.

2.9 MODULAR ARITHMETIC

2.9.1 Def. (Congruent modulo \(n\)) (Skipped)

2.9.7 Notation. (The Set of Congruence Classes Module \(n\))

\[\mathbb{Z/Z}n,\mathbb{Z}/n\mathbb{Z},\mathbb{Z}/(n), \]

2.10 THE CORRESPENDENCE THEOREM

2.10.1 Notation. (Restrict)

Let \(\varphi: G \to \mathcal{G}\) be a group homomorphism, and let \(H \leq G\). We may restrict \(\varphi\) to \(H\), obtaining a homomorphism

\[\varphi |_{H} :H \to \mathcal{G}. \]

We have

\[\ker(\varphi|_{H}) = (\ker \varphi) \cup H. (2.10.2) \]

2.10.4 Prop.

Let \(\varphi:G\to \mathcal{G}\) be a homomorphism, \(K = \ker \varphi\), \(\mathcal{H} \leq \mathcal{G}\). Denote the inverse image \(\varphi^{-1}(\mathcal{H})\) by \(H\). Then \(K \subset H \leq G\).
If \(\mathcal{H} \unlhd \mathcal{G}\), then \(H \unlhd G\).
If \(\varphi\) is surjective and \(H \unlhd G\), then \(\mathcal{H} \unlhd \mathcal{G}\).

2.10.5 Theorem. (Correspondence Theorem)

Let \(\varphi:G \to G'\) be a surjective group homomorphism, \(K = \ker \varphi\), then

\[\{ \text{subgroups of } G \text{ that contains } K \} \leftrightarrow \{ \text{subgroups of }G \} \]

2.11 PRODUCT GROUPS

2.11.1 Def. (Product of Group)

Let \(G\), \(G'\) be groups. The product set \(G\times G' = (a, a') , a\in G, a'\in G'\). The multiplication rule is defined as:

\[(a, a')\cdot(b, b') = (ab, a'b') \]

2.11.3 Prop.

Let \(r\) and \(s\) be relatively prime integers. A cyclic group of order \(rs\) is isomorphic to the product of a cyclic group of order \(r\) and a cyclic group of order \(s\).

2.11.4 Prop. (Product Groups)

Let \(H, K \leq G\), the multiplication map \(f: H\times K \to G\), defined by \(f(h,k)=hk)\). Its image is the set \(HK=\{hk|h\in H, k\in K\}\).

• (a) \(f\) is injective if and only if \(H\cup K = \{1\}\).
• (b) \(f\) is a homomorphism from the product group \(H\times K\) to \(G\) if and only if \(hk=kh\). (It may not be a homomorphism!)
• (c) If \(H\unlhd G\), then \(HK \leq G\).
• (d) \(f\) is an isomorphism from the product group \(H\times K\) to \(G\) if and only if \(H\cup K = \{1\}\), \(HK=G\), and also \(H, K \unlhd G\).

2.11.5 Prop.

There are 2 isomorphism classes of groups of order 4, the class of cyclic group \(C_4\) of order 4 and the class of the Klein Four Group, which is isomorphic to \(C_2\times C_2\).

2.12 QUOTIENT GROUPS

2.12.1 Def. (Quotient Groups)

The set of cosets of a normal subgroup \(N\) of a group \(G\) is often denoted by \(G/N\).

Let \(C \in G/N\), \([C]\) is used to denote the coset. If \(C=aN\), we can denote \(\bar{a} = [C]\), \(\bar{G} = G / N\).

2.12.2 Theorem.

Let \(N \unlhd G\), \(\bar{G} = G/N\). There is a law of composition on \(\bar{G}\) that makes this set into a group, \(\pi : G\to G'\) defined by \(\pi(a)=\bar{a}\) is a surjective homomorphism, and \(\ker \pi = N\).

• The map \(\pi\) is often referred to as the canonical map from \(G\) to \(G'\). 'Canonical' indicates that this is the only map that we might reasonably be talking about.

2.12.3 Corollary.

Let \(N \unlhd G\), and let \(\bar{G} \in G/N\). Let \(\pi : G \to \bar{G}\) be the canonical homomorphism. Let \(a_1,\cdots,a_n\) be elements of \(G\) such that the product \(a_1\cdots a_k \in N\). Then \(\bar{a_1}\cdots \bar{a_k} = \bar{1}\).

2.12.5 Lemma.

Let \(N \unlhd G\), the product \((aN)(bN)\) is also a coset, and it is equal to the coset \(abN\).

This lemma allows us to define multiplication on the set \(\bar{G} = G/N\). If \(C_1\) and \(C_2\) are cosets, then \([C_1][C_2] = [C_1C_2]\).

2.12.8 Lemma.

Let \(G\) be a group, and let \(Y\) be a set with a law of composition. Let \(\varphi: G\to Y\) be a surjective map with the homomorphism property, that \(\varphi(a)\varphi(b) = \varphi(ab), \forall a,b \in G\). Then \(Y\) is a group and \(\varphi\) is a homomorphism.

2.12.10 Theorem. (First Isomorphism Theorem)

Let \(\varphi:G \to G'\) be a surjective group homomorphism with kernel \(N\). The quotient group \(\bar{G} = G/N\) is isomorphic to the image \(G'\). To be precise, let \(\pi:G\to G'\) be the canonical map. There is a unique isomorphism \(\bar{\varphi}: \bar{G} \to G'\) such that \(\varphi = \bar{\varphi} \circ \pi\).