Review
Recall that a group consists of a non-empty set G and a binary operation on G, usually written as multiplication, satisfying the fol- lowing conditions:
• The binary operation is associative: (xy)z = x(yz) for any x,y,z E G
• There is a unique element 1 E G, called the identity element of G, such that xl = x and lx = x for any x E G
In a group G, every element x has a unique inverse x-1, satisfying the properties xx-1 = 1 and x-1x = 1 The associative property allows for the clear calculation of products involving multiple elements, though the order of multiplication is crucial, as xy does not always equal yx When xy = yx, the elements x and y are said to commute The commutator of x and y is defined as [x, y] = xyx-1y-1, indicating that x and y commute if and only if [x, y] = 1.
A group G is defined as abelian if every pair of its elements commutes, making the order of multiplication irrelevant; if not, G is classified as non-abelian In an abelian group, the group operation can be expressed additively, where the product of elements x and y is represented as x + y instead of xy Additionally, the inverse of an element x is indicated by -x, and the identity element is represented by 0.
If x is an element of a group G, then for n E N we use xn
In an abelian group, we define the exponentiation of an element \( x \) as \( x^n \) for a natural number \( n \), with \( x^0 = 1 \) The standard rules of exponentiation apply, and we say that \( x \) has finite order if there exists a positive integer \( n \) such that \( x^n = 1 \) The order of \( x \) is the smallest positive integer \( n \) satisfying this condition, indicating that the elements \( 1, x, x^2, \ldots, x^{n-1} \) are distinct within the group \( G \) and that \( x^n \) equals the identity element.
A group G is classified as finite if it contains a limited number of elements, while it is considered infinite if it does not The order of a finite group G, represented as |G|, refers to the total number of its elements, and similarly, |S| denotes the cardinality of any finite set S In finite groups, every element possesses a finite order, though there exist infinite groups that also exhibit this characteristic, known as periodic groups Conversely, there are infinite groups where only the identity element has a finite order, which are termed torsion-free groups.
A subset H of a group G is said to be a subgroup of G if it forms a group under the restriction to H of the binary operation on G
Equivalently, H ~ G is a subgroup iff the following conditions hold:
• The identity element 1 of G lies in H
• If x, y E H, then their product xy in G lies in H
• If x E H, then its inverse x- 1 in G lies in H
In group theory, every group G is a subgroup of itself, and the trivial subgroup, represented as {1}, is also a subgroup of G While all subgroups of a finite group are finite, infinite groups can have both finite and infinite subgroups, including the trivial subgroup and the group itself In abelian groups, all subgroups are abelian, whereas non-abelian groups contain both abelian and non-abelian subgroups If H is a subgroup of G, it is denoted as H ⊆ G, and if H is a proper subset of G, it is written as H < G Additionally, if K is a subgroup of H and H is a subgroup of G, then K is also a subgroup of G.
PROPOSITION 1 If H and K are subgroups of a group G, then so is their intersection H n K More generally, the intersection of any collection of subgroups of a group is also a subgroup of that group •
The following theorem gives important information about the na- ture of subgroups of a finite group
LAGRANGE'S THEOREM Let G be a finite group, and let H :::;; G Then IHI divides IGI •
If X is a subset of a group G, then we define to be the in- tersection of all subgroups of G which contain X By Proposition 1,
The subgroup of G generated by a subset X, denoted as , is the smallest subgroup of G that includes X, meaning it is contained within any subgroup that also contains X If X is already a subset of G, then equals X For a single element X = {x}, we use to represent the generated subgroup, and for multiple elements X = {x1, , Xn}, we denote it as .
PROPOSITION 2 Let X be a subset of a group G Then consists of the identity and all products of the form x~ 1 • • • x~r where r EN, xi EX, and Ei = ±1 for all i •
A group G is considered cyclic if it can be expressed as G = for some element g in G, where g is referred to as a generator of G For instance, if G is a group of order n and contains an element g with order n, then G can be represented as , consisting of n distinct elements: g, g², , g^(n-1), g^n = 1 According to Proposition 2, the cyclic group can be defined as = {g^n | n ∈ Z}, which demonstrates that cyclic groups are abelian due to their exponentiation properties.
In group theory, cyclic groups are typically represented multiplicatively For an element \( g \) of order \( n \), the cyclic group generated by \( g \) is denoted as \( = \{1, g, \ldots, g^{n-1}\} \), establishing that the order of the group \( || = n \) If \( g \) has infinite order, \( \) forms a torsion-free infinite abelian group Finite cyclic groups of the same order are considered equivalent, as are infinite cyclic groups, with the integers \( \mathbb{Z} \) serving as the canonical example of an infinite cyclic group, and \( \mathbb{Z}/n\mathbb{Z} \) representing the canonical cyclic group of order \( n \) If \( G \) is a finite group and \( g \) is an element of \( G \) with order \( n \), this structure applies accordingly.
In a finite group G, if is a subgroup of order n, Lagrange's theorem indicates that n divides the order of the group |G| This implies that the order of any element in a finite group must also divide the group's order Therefore, if |G| equals a prime number p, each element in G must have a non-trivial divisor of p, leading to the conclusion that G is cyclic Consequently, every non-identity element in G serves as a generator.
In a group G, the product of two subsets X and Y is defined as XY = {xy | x ∈ X, y ∈ Y} This definition can be extended to any finite number of subsets within G Additionally, the inverse of a subset X is defined as X⁻¹ = {x⁻¹ | x ∈ X} A non-empty subset H of G is considered a subgroup if it satisfies the conditions HH = H and H⁻¹ = H.
PROPOSITION 3 Let Hand K be subgroups of a group G Then
HK is a subgroup of G iff HK = KH •
Observe that if Hand K are subgroups of G, then their product
H K contains both H and K; if in addition K :::;; H, then H K = H
(These properties do not hold if Hand K are arbitrary subsets of G.)
If G is abelian, then HK = KH for any subgroups Hand K of G, and hence the product of any two subgroups of an abelian group is a subgroup
We can now describe the subgroup structure of finite cyclic groups
THEOREM 4 Let G = be a cyclic group of order n Then: (i) For each divisor d of n, there is exactly one subgroup of G of order d, namely
If d and e are divisors of n, the intersection of the subgroups with orders d and e results in a subgroup with an order equal to the greatest common divisor (gcd) of d and e Additionally, the product of these subgroups of orders d and e forms a subgroup with an order that corresponds to the least common multiple (lcm) of d and e.
In group theory, when we have a subgroup H of a group G and an element x in G, we denote the left coset of H in G as xH, which represents the set {x}H Conversely, Hx denotes the right coset of H in G In this context, the term "coset" will refer specifically to left cosets throughout this book Although we focus on left cosets, it's important to note that any concept we discuss regarding left cosets has an equivalent in right cosets, as many group theory texts prefer the latter Additionally, there exists a bijective correspondence between left and right cosets, highlighting their interconnected nature.
H in G, sending a left coset xH to its inverse (xH)- 1 = Hx- 1 •
Let H be a subgroup of G Any two cosets of H in G are either equal or disjoint, with cosets xH and yH being equal iff y- 1 x E H
Consequently, an element x E G lies in exactly one coset of H, namely xH For any x E G, there is a bijective correspondence between H and xH; one such correspondence sends h E H to xh
The index of a subgroup H in a group G, denoted IG: HI, represents the number of cosets of H in G In cases where H has an infinite number of cosets in G, IG: HI can be defined as the corresponding cardinal number, while IGI is redefined as the cardinal number IG: 11, ensuring the accuracy of subsequent statements.
In the context of group theory, partitioning a group \( G \) into \( IG \) results in \( HI \) disjoint sets, each with cardinality \( IHI \), leading to the relationship \( IGI = IG : HIIHI \) This observation provides a proof of Lagrange's theorem, although it can also be established through a straightforward counting argument without referencing cosets Notably, all subgroups of a finite group possess a finite index, while subgroups of an infinite group may exhibit either finite or infinite index The collection of cosets, referred to as the coset space of \( H \) in \( G \), is denoted by \( G / H \).
This article provides a comprehensive overview of the subgroups of infinite cyclic groups We encourage readers to rephrase Theorem 4 to highlight the similarities between Theorems 4 and 5 more clearly.
Automorphisms
The set of automorphisms of a group G is referred to as Aut(G) When two automorphisms, rp and p, are composed, their combination rp o p remains an automorphism of G, establishing a binary operation within Aut(G) This operation forms a group structure, where the identity element is the trivial automorphism that maps each element to itself, and the inverse of an automorphism rp is represented by its inverse r.p-1 as a set map Thus, Aut(G) is defined as the automorphism group of G, and we can denote the composition of automorphisms simply as rpp instead of rp o p for rp, p in Aut(G).
In a group G, each element g defines a conjugation homomorphism rp9: G → G, given by rp9(x) = gxg⁻¹ This homomorphism satisfies the properties rp9(xy) = rp9(x)rp9(y) and rp9(x⁻¹) = rp9(x)⁻¹, confirming that it is indeed an automorphism of G The maps rp9 are known as the inner automorphisms of G, and for any elements g, h in G, we have rp9(hxh⁻¹) = rp9(g)rp9(h) for any x in G This leads to a homomorphism from G to Aut(G), mapping each g in G to rp9, with the image referred to as the inner automorphism group Inn(G) and the kernel as the center of G, denoted Z(G).
= {g E G I gx = xg for all x E G}, and hence that Z(G) consists of those elements of G which commute with every element of G Clearly, G is abelian iff Z(G) =G
If \( u \in \text{Aut}(G) \) and \( rp \in \text{Inn}(G) \), it can be verified that \( urp = u^{-1} \) This demonstrates that \( \text{Inn}(G) \) is isomorphic to \( \text{Aut}(G) \), leading to the definition of the outer automorphism group \( \text{Out}(G) = \text{Aut}(G) / \text{Inn}(G) \) The term "outer automorphism" typically refers to automorphisms of \( G \) that are not inner, possessing a non-trivial image in \( \text{Out}(G) \) under the natural mapping In the case of an abelian group \( G \), all non-trivial automorphisms are considered outer, since \( \text{Inn}(G) = 1 \).
When analyzing a group, understanding the structure of its automorphism group can be a challenging task In this article, we will focus specifically on the automorphism groups of cyclic groups, exploring their characteristics and implications in detail.
Let G be a group generated by x, and let t.p be an automorphism of G In this case, t.p(x) must also generate G, with the only possible generators being x and its inverse, x-1 Consequently, t.p either fixes each element or maps each element to its inverse, leading to the conclusion that the automorphism group of G, denoted as Aut(G), is isomorphic to Z2.
Let \( n \) be a natural number and define \( G = \langle x \rangle \cong \mathbb{Z}_n \) If \( t.p \) is an endomorphism of \( G \) such that \( t.p(x) = x^m \) for \( 0 \leq m < n \), then \( t.p \) maps every element of \( G \) to its \( m \)-th power Consequently, \( G \) possesses exactly \( n \) endomorphisms, represented by the \( m \)-th power maps for \( 0 \leq m < n \).
PROPOSITION 1 Let G = ~ Zn for n E N, and for each
0 :::; m < n let am be the endomorphism of G sending x to xm
Then Aut(G) consists precisely of those am for which m =/= 0 and gcd(m,n) = 1 Furthermore, Aut(G) is abelian and is isomorphic with the group (Zjnz)x of units of the ring Z/nZ
The map \( a^0 \) has a trivial image, indicating it is not an automorphism For \( 1 \leq m < n \), if \( \gcd(m, n) = 1 \), integers \( a \) and \( b \) exist such that \( am + bn = 1 \) This leads to the conclusion that \( a^m(X_a) = X^{am} = x^{1-bn} = x(x^n)^{-b} = X \), demonstrating that \( a^m \) is surjective Given that \( G \) is finite, a surjective map from \( G \) to \( G \) must also be injective, confirming \( a^m \in \text{Aut}(G) \) Conversely, if \( a^m \in \text{Aut}(G) \), the properties of the automorphism hold.
X = am(xa) = xam for some a E Z; since xam- 1 = 1, we must have am- 1 = bn for some bE Z, which forces gcd(m, n) = 1 The first assertion now follows
Given 1:::; m1,m2 < n, we have am 1 am 2 =at= am 2 am 1 , where
1 :::; t < n is such that m 1m 2 = t (mod n); therefore Aut(G) is abelian Since (Zjnz)x = {m + nZ 11:::; m < n, gcd(m, n) = 1}, we easily see that the map sending am to m + nZ is an isomorphism from Aut(G) to (Z/nZ)X •
The totient of a positive integer n, denoted as φ(n), represents the count of positive integers less than n that are coprime to n This value is derived from the Euler phi-function and can be expressed as φ(n) = (p₁ - 1)p₂^(k₂ - 1) (pᵣ - 1), where n is factored into distinct prime factors p₁, p₂, , pᵣ Notably, the order of the automorphism group Aut(Zₙ) is equal to φ(n), and specifically, for a prime p, the size of the automorphism group is |Aut(Zₚ)| = p - 1.
PROPOSITION 2 Let p be a prime Then Aut(Zp) ~ Zp-lã
In a field F with p elements, it is established that the automorphism group Aut(Zp) is isomorphic to the multiplicative group Fx of non-zero elements in F For each divisor d of p-1, we define fd as the count of elements in Fx that have order d, while zd represents the number of elements of order d in Zp-1.
Let \( d \) be a divisor of \( p-1 \) If \( x \in F_x \) has an order that divides \( d \), then \( x \) is a root of \( X^d - 1 \in F[X] \), which can have at most \( d \) roots Therefore, if \( x \) has order \( d \), the powers of \( x \) are the only elements in \( F_x \) that are roots of \( X^d - 1 \) This implies that every element of \( F_x \) with order \( d \) must be contained in \( \langle x \rangle \cong \mathbb{Z}_d \) Thus, either \( f_d = 0 \) or \( f_d \) equals the number of elements of order \( d \) in \( \mathbb{Z}_d \).
According to Theorem 1.4, any divisor \( d \) of \( p - 1 \) indicates that all elements of order \( d \) in \( \mathbb{Z}_{p-1} \) are part of a single cyclic subgroup of order \( d \) Consequently, the number of elements of order \( d \) in \( \mathbb{Z}_{d} \) is equal to \( z_d \) This leads to the conclusion that \( z_d \leq d \) for every divisor \( d \) of \( p - 1 \).
L /d = IFXI = p -1 = IZp-11 = L Zd, dj(p-1) dj(p-1) which forces /d = zd for every d I (p- 1) In particular, we have
/p-1 = Zp-1 > 0, and therefore Fx ~ Zp_1 •
Let G = ~ Zn for n ∈ N, and consider the mth power automorphism crm of G, where 1 ≤ m ≤ n and gcd(m,n) = 1 A straightforward induction shows that (crm)k(x) = x^mk for any k ∈ N Consequently, the order of crm is defined as the smallest positive integer k such that x^mk = x, or equivalently, the smallest k ∈ N satisfying mk ≡ 1 (mod n) If the order of crm equals the totient of n, m is termed a primitive root modulo n, a concept rooted in classical number theory Notably, Aut(Zn) is cyclic if and only if a primitive root modulo n exists.
For composite integers n, understanding the structure of Aut(Zn) primarily involves number theory rather than group theory A notable result characterizes the specific values of n for which Aut(Zn) is cyclic.
THEOREM 3 Aut(Zn) is cyclic iff n = 2 or 4, or n = pk or 2pk for some odd prime p and some k E N •
A proof of the equivalent result about the existence and non-existence of primitive roots modulo n is given in [9, Section 8.3]
Let \( \varphi \) be an automorphism of a group \( G \), and let \( H \) be a subgroup of \( G \) The automorphism \( \varphi \) maps \( H \) isomorphically to a subgroup \( \varphi(H) \) of \( G \), and we say that \( H \) is fixed by \( \varphi \) if \( \varphi(H) = H \) In this scenario, the restriction of \( \varphi \) to \( H \) is an automorphism of \( H \) If \( L \) is a subgroup of \( \text{Aut}(G) \), then \( H \) is fixed by \( L \) if it is fixed by every \( \varphi \in L \) This leads to the conclusion that \( H \) is normal in \( G \) if and only if \( H \) is fixed by \( \text{Inn}(G) \) Additionally, we define \( H \) as a characteristic subgroup of \( G \).
A subgroup H is considered characteristic in a group G if it remains unchanged under all automorphisms of G, denoted as H char G An example of this is the center Z(G), which is always a characteristic subgroup of G This is demonstrated by the fact that if an element x belongs to Z(G) and φ is an automorphism of G, then φ(x)y = φ(xφ⁻¹(y)) = φ(φ⁻¹(y)x) = yφ(x) for any y in G, confirming that φ(x) is also in Z(G) While all characteristic subgroups are normal, the reverse is not necessarily true; for instance, an infinite abelian group may lack any non-trivial proper characteristic subgroups.
We observed in Section 1 that being normal is not a transitive property of subgroups However, being characteristic is transitive:
LEMMA 4 If K is a characteristic subgroup of H and H is a characteristic subgroup of G, then K is a characteristic subgroup of G
PROOF If