Basic Definition of a Group#
A group is an algebraic structure consisting of a nonempty set \(G\) together with a binary operation (denoted by \(\circ\)) on it, satisfying the following axioms:
Associativity: \(\forall a, b, c \in G\),
\[ (a \circ b) \circ c = a \circ (b \circ c). \]Identity element: There exists an element \(e \in G\) such that for every \(a \in G\):
\[ e \circ a = a \circ e = a. \]Inverse element: For each \(a \in G\), there exists an element \(a^{-1} \in G\) such that
\[ a \circ a^{-1} = a^{-1} \circ a = e. \]
According to this definition, the elements of \(G\) are closed under the operation \(\circ\).
It is important to note that the set \(G\) itself and the group \(G\) are two distinct mathematical objects, although they often share the same symbol. A group has an operation structure that goes beyond just a set. In more formal contexts, it is denoted \((G, \circ)\). We call the set on which the group is based the underlying set of the group \(G\).
The two most common binary operations are addition and multiplication. A group only defines one operation, which leads to the concept of an additive group and a multiplicative group. In an additive group, the identity element is 0, and the inverse of any element \(x\) is \(-x\). In a multiplicative group, the identity element is 1, and the inverse of any element \(x\) is \(x^{-1}\). Moreover, operations such as permutation composition, matrix multiplication, symmetric difference, XOR, and function composition can all form groups.
The definition of a group does not require the operation to be commutative. Accordingly, we can distinguish between Abelian groups and non-Abelian groups.
The order of a group, denoted \(|G|\), is defined as the number of elements in the group. A group is called a finite group if it has a finite number of elements (finite order), and an infinite group if it has infinitely many elements (infinite order).
Basic Concepts#
When studying sets, we use concepts such as subsets, functions, and equivalence relations and their quotients. Similarly, when studying groups, we use the concepts of subgroups, homomorphisms, and quotient groups. These are introduced below.
Subgroups#
If \(H\) is a nonempty subset of \(G\), and \(H\) also forms a group under the operation \(\circ\) inherited from \(G\), then \(H\) is called a subgroup of \(G\). We denote this by \(H \leq G\).
The conditions for a subgroup can be simplified to the following criteria:
- Closure: \(\forall a, b \in H\), we have \(a \circ b \in H\).
- Identity element: The identity element \(e\) of the group \(G\) is in \(H\).
- Inverse element: \(\forall a \in H\), its inverse \(a^{-1}\) is also in \(H\).
If \(H \neq G\) and \(H \leq G\), then \(H\) is called a proper subgroup of \(G\), denoted \(H < G\).
Cosets#
Let \(H\) be a subgroup of \(G\). Then \(H\) partitions the elements of \(G\) into several disjoint subsets of equal size, called the cosets of \(H\). Cosets can be divided into left cosets and right cosets. Given an element \(g \in G\):
- Left coset: \(gH = {g \circ h \mid h \in H}\)
- Right coset: \(Hg = {h \circ g \mid h \in H}\)
It is easy to see that each coset of \(H\) has the same order as \(H\), and \(H\) itself is both a left coset and a right coset. The number of left cosets is equal to the number of right cosets, called the index of \(H\) in \(G\), denoted \([G : H]\).
An important theorem is Lagrange’s Theorem: if \(H\) is a subgroup of \(G\), then \(|G| = |H|/ [G : H]\). This theorem describes the relationship between the order of a group and the order of its subgroup. In particular, if \(G\) is a finite group, we get the corollary that only numbers that divide \(|G|\) can be the order of a subgroup, hence any subgroup of prime order is necessarily a cyclic group. Also, the order of every element of \(G\) divides the order of \(G\).
Normal Subgroups#
If for every \(g \in G\), the left coset and the right coset coincide, i.e., \(gH = Hg\), then \(H\) is called a normal subgroup of \(G\), denoted \(H \triangleleft G\). An equivalent definition is that a normal subgroup \(H\) is invariant under the conjugation action of any element \(g \in G\), i.e., \(g H g^{-1} = H\).
Quotient Groups#
On the set of cosets, one can define a binary operation that satisfies the group axioms, thereby forming a quotient group. Formally, if \(H\) is a normal subgroup of \(G\), then on the set of all left cosets (or right cosets) of \(H\) in \(G\), we define
$$
(aH) \circ (bH) = (a \circ b)H.
$$
This defines a group structure called the quotient group, denoted \(G/H\).
In fact, it can be shown that for this quotient group structure to be well-defined, \(H\) must be a normal subgroup. When left and right cosets are not equal, one cannot define a binary operation that satisfies the group axioms (though one can still define related structures, forming a homogeneous space).
The reason it is called a “quotient” group is analogous to integer division: for example, dividing 8 by 2 to get 4 is equivalent to partitioning 8 objects into two subsets each containing 4 objects. In a similar manner, forming the quotient group corresponds to partitioning the group \(G\) by \(H\).
Homomorphisms and Isomorphisms#
A group homomorphism is the fundamental tool for studying the relationships between different groups. Let \(G\) and \(H\) be two groups. A map \(\phi: G \rightarrow H\) is called a homomorphism from \(G\) to \(H\) if, for all \(a, b \in G\), it satisfies $$ \phi(a \circ_G b) = \phi(a) \circ_H \phi(b). $$
A homomorphism preserves the group operation and has the following basic properties:
- The identity element maps to the identity element: \(\phi(e_G) = e_H\).
- Inverse elements map to inverse elements: \(\phi(a^{-1}) = \phi(a)^{-1}\).
An isomorphism is a special type of homomorphism. If \(\phi: G \rightarrow H\) is a bijective (both injective and surjective) homomorphism, then \(\phi\) is called an isomorphism, denoted \(G \cong H\). Two isomorphic groups are structurally the same, differing only in how their elements are represented.
When studying homomorphisms, two important concepts are:
Kernel: For a homomorphism \(\phi: G \rightarrow H\), the kernel is the set of all elements in \(G\) that map to the identity element in \(H\): \[ \ker(\phi) = {g \in G \mid \phi(g) = e_H}. \]
Image: The image of \(\phi\) is the subset of \(H\) consisting of all elements that come from some \(g \in G\) under \(\phi\): \[ \mathrm{im}(\phi) = {\phi(g) \mid g \in G}. \]
It can be shown that the kernel of a homomorphism is a normal subgroup of the original group, and the image is a subgroup of the target group.
One of the most important results in the theory of homomorphisms is the First Isomorphism Theorem: if \(\phi: G \rightarrow H\) is a group homomorphism, then $$ G/\ker(\phi) \cong \mathrm{im}(\phi). $$ This theorem reveals the essential connection among the group, its kernel, the quotient group, and the image, and it provides a powerful tool for exploring the structure of groups.
Example#
Imagine a square sheet of paper lying on a plane. Consider all the geometric transformations (such as rotations and reflections) that map the square onto itself. These transformations form a group. We denote it by \(D_4\), commonly called the “dihedral group of the square.” Its underlying set contains eight elements (eight rigid motions), and the operation is “performing transformations in sequence” (i.e., composition). For convenience of description, we denote a clockwise rotation by \(90^\circ\) as \(r\), and a reflection about a fixed axis as \(s\). The eight elements of \(D_4\) can be written as: $$ \{e,\; r,\; r^2,\; r^3,\; s,\; rs,\; r^2s,\; r^3s \}, $$ where \(e\) denotes the “do nothing” or identity transformation; \(r^k\) denotes a clockwise rotation by \(90^\circ\) applied \(k\) times; \(s\) represents a single reflection; \(r^k s\) represents the composition of a reflection and \(k\) rotations (or vice versa, depending on the convention, but conceptually it is “rotation + reflection”).
In \(D_4\), any composition of three transformations can be reduced to a composition of two transformations by associativity, avoiding ambiguity. The identity element \(e\) satisfies \(e \circ g = g \circ e = g\) for any element \(g\). Every transformation has an “inverse,” for example, the inverse of \(r\) is \(r^3\), and the inverse of a reflection \(s\) is itself \(s\). Thus, \(D_4\) is indeed a group.
Now consider a subgroup. There is a notable subgroup in \(D_4\) consisting of only the rotations: \(\langle r\rangle = \{e, r, r^2, r^3\}\). It has four elements, and it meets the three subgroup criteria mentioned earlier, so \(\langle r\rangle\) is a subgroup of \(D_4\). As a subgroup, \(\langle r\rangle\) shares the same identity element \(e\) with \(D_4\), and the inverse elements in \(\langle r\rangle\) coincide with their inverses in \(D_4\). Moreover, \(\langle r\rangle\) itself is a cyclic group.
If we use \(\langle r\rangle\) to form cosets, \(D_4\) can be partitioned into two disjoint subsets of equal size: \(\langle r\rangle\) itself (which is both a left and a right coset), and \(s \langle r\rangle = \{s, rs, r^2s, r^3s\}\). We can get an intuitive sense of this partition: any element that lies in the first subset involves “pure rotation,” and any element that lies in the second subset involves “reflection.” These two subsets do not overlap, and they together partition all eight elements of \(D_4\). This shows that the index \([D_4 : \langle r\rangle]\) is 2, and verifies Lagrange’s theorem with \(|D_4| = 8\) and \(|\langle r\rangle| = 4\): \(8 = 4 \times 2\).
Further observation reveals that \(\langle r\rangle\) is also a normal subgroup of \(D_4\), namely for every \(g \in D_4\), \(g \langle r\rangle g^{-1} = \langle r\rangle\). Intuitively, “rotating a full turn” and “unrotating,” or “reflecting and unreflecting,” keeps one within the pure rotation subgroup. Since \(\langle r\rangle\) is a normal subgroup, we can define a group operation on its cosets, forming the quotient group \(D_4 / \langle r\rangle\). This quotient group has only two elements: one is \(\langle r\rangle\) itself, and the other is \(s \langle r\rangle\). The operation is defined as $$ (a\langle r\rangle) \circ (b\langle r\rangle) = (a \circ b)\langle r\rangle. $$ It may look somewhat abstract, but essentially, you can think of these two cosets as two types of symmetry: “pure rotation” versus “reflection.” The entire structure matches that of a group with only two elements (often denoted \(\mathbb{Z}_2\) or \({0,1}\)), meaning that on a high level, we treat “pure rotation” and “reflection” as two distinct equivalence classes.
Finally, define a map \(\phi: D_4 \to \{1, -1\}\) (where \(\{1, -1\}\) can be viewed as a two-element group under multiplication): let every “pure rotation” element map to \(1\), and every element involving “reflection” map to \(-1\). This preserves the group structure: composing two pure rotations remains a pure rotation (\(1 \times 1 = 1\)), composing a rotation with a reflection yields a reflection (\(1 \times -1 = -1\)), and composing two reflections brings you back to a rotation (\((-1) \times (-1) = 1\)). Thus, \(\phi\) is a group homomorphism. Its kernel is precisely \(\langle r\rangle\), consisting of all elements that map to \(1\); its image is \(\{1, -1\}\), which is the entire target group. By the First Isomorphism Theorem, we obtain $$ D_4 / \langle r\rangle \cong \{1, -1\}, $$ which corresponds to our earlier understanding of the quotient group.