2. Groups

Groups

The group is a notion even more primitive than the ring, because in case of a group we will be only considering one operation (depending on the context it will be denoted by \cdot or +). So a set G with a fixed neutral element 1 and with operation \cdot is a group if

  • for any a,b,c\in G, (a\cdot b)\cdot c= a\cdot(b\cdot c),
  • for any a\in G, a\cdot 1 = 1\cdot a = a,
  • for any a\in G there exists b\in G (denoted by a^{-1}) such that a\cdot b= b\cdot a =1.

If additionally the operation \cdot is commutative, i.e. for any a,b\in G, ab=ba, then such a group is called an abelian group. Actually, in this case + is usually used to denote the operation (thus we use 0 instead of 1, and -a instead of a^{-1} then).

Notice that if P is a commutative ring with an identity, it is an abelian group with its operation + (and neutral element 0) as well, which is called the additive group of this ring. Also the set of units of a ring P, denoted by U(P) is an abelian group with operation \cdot and neutral element 1. This group is called the multiplicative group of P.

The set of all matrices n\times n over a field K with operation of multiplication and neutral element I, denoted by GL(n,K), is a group. Notice that this group is not abelian.

Subgroups

A subset H\subseteq G is a subgroup if 1\in H and for g,h\in H also gh\in H and g^{-1}\in H. We denote this by H\leq G.

E.g. the set of all matrices n\times n over K with determinant 1 is a subgroup of GL(n,K). It is denoted by SL(n,K). Thus, SL(n,K)\leq GL(n,K).

Homomorphisms

A mapping \varphi\colon G\to H, where G,H are groups, is a homomorphism, if for any g,g'\in G, \varphi (gg')=\varphi(g)\varphi(g'). It follows that \varphi(1)=1 and \varphi(g^{-1})=(\varphi(g))^{-1}.

Notice also that \text{im}\varphi=\varphi[G]\leq H and the kernel \ker \varphi=\varphi^{-1}[\{1\}]\leq G.

E.g. \varphi\colon \mathbb{Z}\to \mathbb{Z}_n, where \varphi(x) is x modulo n is a homomorphism between the additive groups of those rings. Also, \det\colon GL(n,K)\to K is yet another example of a homomorphism.

Order of an element and of a group

The order of a group G is its cardinality |G| is it is finite. In the other case we say that the order of the group is infinite.

The order of an element g\in G, o(g), is the least natural number n, such that g^n=1, or \infty, is such n does not exist.

E.g. o(2)=3 is the additive group of \mathbb{Z}_6, since 2+2+2=0 (zero is the neutral element, and + is the operation here).

A group is a torsion group if all its elements are of finite order.

Cosets of a subgroup

If H\leq G, and g\in G, then gH=\{gh\colon h\in H\} is a left coset of H. Notice that gH=g'H if and only if g^{-1}g\in H.

The set of all left cosets is denoted by G/H. Its cardinality is called the index of H in G and is denoted by [G:H]. Lagrange’s Theorem states that in every finite group G, with H\leq G, we have |G|=|H|\cdot [G:H].

Normal subgroups

A subgroup H\leq G is normal, if for any g\in G, the set gHg^{-1}=\{ghg^{-1}\colon h\in H\}=H. This is denoted by H\trianglelefteq G.

It is important to notice that for every homomorphism its kernel is a normal subgroup.

Generators, cyclic groups

We say that a subgroup H\leq G is generated by a set X\subseteq G, if H is the least subgroup containing X, which is denoted by H=\langle X\rangle. In particular, then (if X\neq \varnothing),

    \[\langle X\rangle = \{g_1^{i_1}\cdot\ldots\cdot g_k^{i_k}\colon k\in \mathbb{\N}, i_j=\pm 1, g_j\in X\}.\]

If G=\langle \rangle, then X is the set of generators of the group G. If X=\{x\}, we simply write G=\langle x\rangle and then the group G is called a cyclic group.

Notice that all subgroups of a cyclic group are cyclic, and if an order of the group is finite, there is exactly one subgroup of order equal to a divisor of the order of the whole group.

Groups of permutations

A permutation of a given set is a bijection from this set onto itself, which can be also understood as an ordering of its elements in the case of finite sets. The set of all permutations of a set A is denoted by \Sigma_A. Permutations can be composed, so along with operation \circ it is a group.

In particular, we will take interest in permutations of the set \{1,2,\ldots,n\}. The set of such permutations will be denoted by \Sigma_n. E.g. the function \sigma(1)=2, \sigma(2)=1, \sigma(3)=3 is an element of \Sigma_3. We will denote the permutations as two rows of numbers, in the upper row the arguments and in the lower the values corresponding to them. Thus the considered above permutation can be denoted as:

    \[\sigma=\left(\begin{array}{ccc}1&2&3\\2&1&3\end{array}\right).\]

Thus, we have for example:

    \[\left(\begin{array}{ccc}1&2&3\\2&1&3\end{array}\right)\circ \left(\begin{array}{ccc}1&2&3\\1&3&2\end{array}\right)=\left(\begin{array}{ccc}1&2&3\\1&3&2\\2&3&1\end{array}\right)=\left(\begin{array}{ccc}1&2&3\\2&3&1\end{array}\right).\]

A permutation \sigma of a set A is a cycle if there are a_1,\ldots, a_n\in A such that \sigma(a_1)=a_2,\sigma(a_2)=a_3,\ldots, \sigma(a_{n-1})=a_n,\sigma(a_n)=a_1, and \sigma is constant on all other elements. Such a permutation will be denoted simply as \sigma=(a_1,a_2,\ldots, a_n). Thus, in the above example, \sigma=(1,2).

We can easily notice that any permutation can be decomposed into an composition of disjoint cycles, e.g.:

    \[\left(\begin{array}{cccccccc}1&2&3&4&5&6&7&8\\6&7&4&1&3&5&2&8\end{array}\right)=(1,6,5,3,4)(2,7)(8).\]

Notice also that the order of each cycle is equal to its length, so the order of each permutation is the leach common multiple of the length of cycles in its decomposition into disjoint cycles.

Finally, each cycle can be decomposed into (not disjoint) transpositions (cycles of length 2). Indeed, (a_1,\dots, a_n)=(a_1,a_n)\circ(a_1,a_2).
One can prove, that although there are multiple possibilities of decomposition of a permutation into transpositions, for a given permutation all the decomposition have either even or odd number of transpositions. Thus, a permutation is even if it can be written as a composition of an even number of transpositions, and otherwise it is odd. The set of all even permutations A_n is a normal subgroup of all permutations \Sigma_n, which can be checked quite easily.

Cayley Theorem states that every group G is isomorphic to a subgroup of the group \Sigma_G. Indeed, every element g\in G can be identified with a permutation of the group G of the form x\mapsto g\cdot x.

Quotient groups

If H\trianglelefteq G is a normal subgroup, then the set G/H of posets can be regarded also as a group with operation gH\cdot g'H= (gg')\cdot H and (gH)^{-1}=g^{-1}H. This makes sense only if H is a normal subgroup, because only then we can be sure that the operation is well-defined, by which we mean that if xH=x'H and yH=y'H, then xyH=x'y'H.

There is a natural homomorphism, which is onto, \pi\colon G\to G/H defined as \pi(g)=gH. Obviously, \ker \pi=H.

Moreover, the homomorhpism theorem states that if \varphi\colon G\to H is a homomorphism, then there exists exactly one monomorphism \varphi'\colon G/\ker\varphi\to H, such that \varphi'\circ \pi=\varphi, where \pi\colon G\to G/\ker \varphi is the projection defined above. In particular, the groups G/\ker\varphi and \text{im}\varphi are isomorphic.

Direct products and sums

Given groups G_1,G_2,\ldots (finitely or infinitely many, i\in I), the Cartesian product of these groups \prod G_{i} is the Catesian product equipped by the coordinatewise operation. Its subgroup is the so called direct product \coprod_{i\in I} G_i=\{(g_i)\colon \exists_n\forall_{i>n} g_i=1\} of all elements with finitely many non-neutral coordinates.

For groups with notation + we use the term direct sum and write \bigoplus_{i\in I} G_i.

On can notice that if A is a group and G_i is a family of groups with G_i\simeq A_i\trianglelefteq A, and A=\langle A_i\colon i\in I\rangle, and also A_i\cap A_j=\{1\} for any i\neq j, then A\simeq \cprod_{i\in I} G_i. In such case we also say that A is the inner product (or inner sum in the case of groups with +) of its subgroups A_i, i.e. A=\coprod_{i\in I} A_{i}.

It is worth noticing that \mathbb{Z}_n=\mathbb{Z}_{p_{1}^{k_1}}\oplus \ldots \oplus \mathbb{Z}_{p_{m}^{k_m}}, where n=p_1^{k_1}\cdot \ldots \cdot p_m^{k_m} is a factorization of n. Actually, it is true for any finite abelian group G in the following sense, G=P_1\oplus\ldots\oplus P_m, where |P_i|=p_i^{k_i} and n=|G|=p_1^{k_1}\cdot \ldots \cdot p_m^{k_m}. Finally, every abelian group of order p^k, where p is a prime number can be described as a direct sum of some of its cyclic subgroups. This means that every finite abelian group is a direct sum of some of its cyclic subgroups.

Centre, commutator, conjugate elements

The set called the centre of the group Z(G)=\{g\in G\colon \forall_{h\in G} gh=hg\} is a normal subgroup of G, which “measures” how much G is commutative. On the other hand let [x,y]=x^{-1}y^{-1}xy, and [G,G]=\langle [x,y]\colon x,y\in G\rangle is a commutator of G. It is also a normal subgroup of G and it measures how G is non-commutative. In particular, a homomorphic image of G is abelian if and only if the kernel contains [G,G].

The element y=g^{-1}xg is called an adjunct of x and the set of all adjuncts of x will be denoted by C_x.

Notice that |C_x|=[G:Z(\{x\})], where Z(\{x\})=\{g\in G\colon gx=xg\}, which means that in the case of finite groups the |C_x| is a divisor of |G|, and obviously since G is a union of classes of adjuncts their cardinalities sum up to the order of G.

Notice also that o(x)=o(g^{-1}xg). Even more can be said in the case of permutations. Namely, \alpha,\beta\in \Sigma_n are conjuncts if and only if their decompositions into disjoint cycles contain the same number of cycles of the same lenght. E.g. \alpha=(1,2)(3,4)(5,6,7) and \beta=(4,6)(5,2)(1,3,7) are conjuncts, because for

    \[\gamma=\left(\begin{array}{ccccccc}1&2&3&4&5&6&7\\4&6&5&2&1&3&7\end{array}\right)\]

we get \beta=\gamma^{-1}\circ\alpha\circ \gamma.

Group action

We say that a group G acts on a set X, if there exists a homomorphism \phi\colon G\to \Sigma_X. Usually we write g(x) instead of (\phi(g))(x). Cayley’s Theorem describes an action of a group on itself. Another obvious example is the action of D_n on the set of verteces of n-polygon.

For x\in X, the set G(x)=\{g(x)\colon g\in G\} is called an orbit of x. Obviously orbits give a partition of X. The set X^G=\{\forall_{g\in G} g(x)=x\} is the set of fixed points. Notice that for every x\in X^g, G(x)=\{x\}. Finally G_x=\{g\in G\colon g(x)=x\} is a subgroup of G, and is the group of stabilizers of x.

It is crucial to notice that |G(x)|=[G:G_x]. In particular this means that the cardinality of every orbit is a divisor of the order of G. This gives |X| as a sum of divisors of G.

An important collorary follows from consideration of group actions, which is Cauchy Theorem. This theorem states that if G is a finite group and a prime number p is such that p||G|, then there exists g\in G such that o(g)=p.