Notes on Huppert

The nonempty set \(G\) is called a Group if it satisfies the following:

1. For each pair \((A,B)\) of elements \(A,B\in G\), there exists a unique element \(C\) of \(G\), which we write as \(C=AB\). We call \(C\) the product of \(A\) and \(B\).
2. For every \(A,B,C\in G\), we have the product be associative: \((AB)C=A(BC)\).
3. There is an element \(E\in G\) such that \(EA=A\) for every \(A\in G\).
4. For every \(A\in G\) there is a \(B\in G\) such that \(BA=E\).

For any non-empty subsets of a group \(G\), we can multiply them together. We recursively define the product \(P_{k}(-)\) — if \(A_{1}\), \(A_{2}\), … are elements of \(G\), and the subscript on \(P\) indicates how many elements we're multiplying together — by:

1. \(P_{1}(A_{1})= \{A_{1}\}\)
2. \(P_{2}(A_{1},A_{2})=\{A_{1}A_{2}\}\)
3. \(P_{n}(A_{1},\dots,A_{h})=\{XY\mid X\in P_{k}(A_{1},\dots,A_{k}), Y\in P_{m}(A_{k+1},\dots,A_{k+m}), k+m=n\}\)

We usually write \(A_{1}\cdots A_{k}\) for \(P_{k}(A_{1},\dots,A_{k})\).

1. \(E\) is the identity element of the group \(G\), so \(AE=A\) for each \(A\in G\)
2. If \(BA=E\), then \(AB=E\)
3. If \(AX=AY\) or \(XA=YA\), then \(X=Y\).

If \(G\) is a group, suppose \(A,B\in G\), then there exists an element \(X\in G\) such that \(AX=B\). There also exists an element \(Y\in G\) such that \(YA=B\).

1. For any \(A\in G\) there is a unique element \(X\in G\) such that \(AX=XA=E\). We call \(X\) the inverse element of \(A\) and denote \(X=A^{-1}\).
2. We have for any \(A,B\in G\) the inverse of their product \((AB)^{-1}=B^{-1}A^{-1}\) and \((A^{-1})^{-1}=A\).

We can define the \(n^{\text{th}}\) power of a group element \(A\in G\) by \(A^{0}=E\), \(A^{n+1}=A^{n}A\) if \(n\gt 0\), and \(A^{-n}=(A^{n})^{-1}\). We also have \(A^{m+n}=A^{m}A^{n}\) for any \(m,n\in\mathbb{Z}\).

We denote, for any set \(M\), its cardinality by \(|M|\). Then we define the Order of a group \(G\) to be the cardinality of its underlying set, and denote it by \(|G|\).

Two groups, \(G_{1}\) and \(G_{2}\), are called isomorphic if there are there is a one-to-one mapping \(\pi\) from \(G_{1}\) to \(G_{2}\)

\(\pi(gh) = \pi(g)\pi(h)\)

for all elements \(g\) and \(h\) from \(G_{1}\). We then write \(G_{1}\cong G_{2}\). Group isomorphism is obviously an equivalence relation.

By Group Property we understand a property which if \(G\) has it, then every group isomorphic to \(G\) also has the property.

If \(\pi\colon G_{1}\to G_{2}\) is an isomorphism and \(E_{1}\in G_{1}\) and \(E_{2}\in G_{2}\) are identity element, then \(\pi(E_{1})=E_{2}\).

Further, for any \(g\in G_{1}\) we have \(\pi(g^{-1})=[\pi(g)]^{-1}\).

The group \(G\) is called Abelian (or also commutative), if \(AB = BA\) for all elements \(A\) and \(B\) from \(G\). Then each product \(A_{1}\dots A_{n}\) with \(A_{i}\in G\) independent of the arrangement of the $A_i.

We call a subset \(U\subset G\) a Subgroup of \(G\) if it is closed under \(G\)'s binary operator:

For any \(u_{1}\), \(u_{2}\in U\) we have \(u_{1}u_{2}\in U\); for any \(u\in U\), we have \(u^{-1}\in U\). We also have \(E\in U\), but this follows by combining these two conditions together.

We write \(U\leq G\) to indicate \(U\) is a subgroup of \(G\). If \(U\neq G\), then we write \(U\lt G\).

Every group \(G\) has a subgroup \(\mathbf{1}=\{E\}\) consisting of only the identity element. We call \(\mathbf{1}\) the Trivial Subgroup of \(G\).

If \(U\) is a finite subset of the group \(G\), and if for any \(u_{1}\), \(u_{2}\), from \(U\) we also have \(u_{1}u_{2}\in U\), then \(U\) is a subgroup of \(G\).

If \(U_{i}\) is a family of subgroups of \(G\) indexed by \(i\in I\), then \(\bigcap_{i\in I}U_{i}\) is a subgroup of \(G\).

Let \(M\) be a subset of a group \(G\). We denote by \(\langle M\rangle\) the intersection of all subgroups of \(G\) which contain \(M\). Obviously \(\langle M\rangle\) is the smallest subgroup of \(G\) containing \(M\).

If we write \(M^{-1}=\{g^{-1}\mid g\in M\}\), then we can write

If \(M=\{A,B,\dots\}\), then we can write \(\langle M\rangle=\langle A,B,\dots\rangle\).

If \(G=\langle M\rangle\), then we call \(M\) a set of Generating Set for \(G\).

We call a group generated by a single element Cyclic. If we always write \(A^{0}=E\), then \(\langle A\rangle\) consists of the elements \(A^{m}\) with \(m=0,\pm1,\pm2,\dots\) which are not necessarily all different.

If \(U\leq G\) is a subgroup, then \(gU=\{gu\mid u\in U\}\) for any \(g\in G\) which we call Left Cosets of \(U\) in \(G\). For any \(g,h\in G\) we have either \(gU\cap hU=\emptyset\) or \(gU=hU\).

The group \(G\) is the disjoint union of all different left cosets \(gU\).

If the set \(\mathcal{R}\) contains exactly one element from each left coset of \(U\), then we call \(\mathcal{R}\) a system of Left Coset Representatives of \(U\) in \(G\).

The decomposition

we call the "decomposition of \(G\) into left cosets by \(U\)".

All cosets of \(U\) have the same cardinality \(|U|\). We call the cardinality \(|\mathcal{R}|\) the Index of \(U\) in \(G\) and write \([G:U]\).

1. Let \(U\) be a subgroup of \(G\). If \(G=\bigcup_{R\in\mathcal{R}}UR\) is the decomposition of \(G\) into right cosets of \(U\), then \(G=\bigcup_{R\in\mathcal{R}}R^{-1}U\) is a decomposition into left cosets of \(U\). (Specifically, we don't have between distinguish a left and right index of \(U\) in \(G\).)
2. If \(G=\bigcup_{R\in \mathcal{R}}UR\) is a decomposition of \(G\) into right cosets of the subgroup \(U\), and \(U=\bigcup_{S\in\mathcal{S}}BS\) is a decomposition of \(U\) by \(B\), then \(G=\bigcup_{S\in\mathcal{S},R\in \mathcal{R}} BSR\) is the decomposition of \(G\) into right-cosets of \(B\). In particular, \([G:B] = [G:U][U:B]\).

Let \(G\) be a finite group, let \(H\leq G\) be a subgroup. Then \(|G|=|H|[G:H]\). In particular, the order and index of a subgroup of a finite group always divide the group's order.

Let \(g\) be an element of group \(G\). The smallest number \(n\in\NN\) with \(g^{n}=E\) we call the Order of \(g\), denoted \(o(g)\). If there is no such number, then \(o(g)=\infty\).

The least common multiple of the orders of all elements \(g\in G\) is the Exponent of \(G\), and denote this by \(\exp(G)\).

Let G be a group. Let \(g\in G\) with \(o(g) = n\lt\infty\). Then:

1. \(n|m\) follows from \(g^{m} = E\).
2. \(\langle g\rangle = \{E, g, g^{2}, \dots, g^{n-1}\}\) and \(\langle g\rangle = o(g) = n\). In particular, \(o(g)\) divides \(|G|\) for finite \(G\).
3. If k is an integer rational number, then \(o(g^{k}) = n/(n,k)\).

Every group of prime order is cyclic.

(This is only the first in a long line of theorems where we derive information about \(G\) from prime factorizations of \(|G|\).)

If \(A\) and \(B\) are subsets of the group \(G\), then we set \(AB=\{ab\mid a\in A,b\in B\}\).

1. If \(A = \bigcup_{R\in \mathcal{R}} R(A\cap B)\) is the coset decomposition from A to \(A\cap B\), then \(AB = \bigcup_{R\in \mathcal{R}} RB\) is a disjoint decomposition of the set \(AB\). In particular, if \(A\) and \(B\) are finite, then \(|AB|=|A||B|/|A\cap B|\).
2. \(AB\) is a subgroup of \(G\) if and only if \(AB = BA\).
3. (Dedehind identity) \(A\leq C\leq G\) and \(C\subset AB\), then it follows that \(C=AB\cap C=A(B\cap C)\).

Let \(A\) and \(B\) be subgroups of \(G\) of finite Index. Then:

1. \([G: A\cap B] \leq [G : A][G : B]\).
2. If \([G: A]\) and \([G: B]\) are divisors, then \([G:A\cap B]=[G:A][G:B]\). If \(G\) is finite, then \(G = AB\).

If \(a\) and \(g\) are elements of \(G\), then we let \(g^{a}=a^{-1}ga\). We call the mapping \(g\to g^{a}\) the transformation with \(a\). then the following rules apply:

• (1) \(g^{ab}=(g^{a})^{b}\).
• (2) \((gh)^{a}=g^{a}h^{a}\)
• (3) \((g^{a})^{-1} = (g^{-1})^{a}\)

We also define \(g^{a+b}\) by \(g^{a+b}=g^{a}g^{b}\). However, it should be noted that then i. a. \(g^{a+b}\neq g^{b+a}\) because the elements \(g^{a}\) and \(g^{b}\) are not necessary interchangeable. But the following always applies:

• (4) \(g^{ab+ac} = (g^{a})^{b+c}\) and
• (5) \(g^{ba + ca} = (g^{b+c})^{a}\).

If \(M\) is a nonempty subset of \(G\) and \(g\in G\), so we set \(M^{g} = \{m^{g}\mid m\in M\}\). We call the sets \(M^{g}\) the conjugates of \(M\) in \(G\).

If \(U\) is a subgroup of \(G\), then \(U^{g}\) is a subgroup of \(G\) isomorphic to \(U\): The group property of \(U^{g}\) follows from 2.14 (2) and (3). The mapping \(\pi\) from \(U\) to \(U^{g}\) with \(\pi(u)= u^{g}\), which is divided by \(\alpha\) with \(\alpha(u^{g}) = (u^{g})^{g^{-1}}\) is reversed, by 2.14 (2) there is an isomorphism from \(U\) to \(U^{g}\).

Let \(M=\{m\}\) be a set of one element. Then we call the set \(\{m^{g}\mid g\in G\}\) of elements conjugate to \(m\) in \(G\) the Conjugacy Class \(m\) in \(G\). The conjugacy classes are equivalence classes for the following equivalence relation:

Let \(x\sim y\) if there is a \(g\in G\) such that \(x^{g} = y\). If \(G\) is finite, then the partition into conjugacy classes yields a disjoint decomposition \(G = \bigcup^{h}_{i}R_{i}\) and the equation

where the \(R_{i}\) are the different conjugacy classes of \(G\). We call the number \(h\) of conjugacy classes the Class Number of \(G\).

1. Let \(M\) be a subset of \(G\). Then we call \(N_{G}(M) = \{g|g\in G, M^{g}=M\}\) the Normalizer of \(M\) in \(G\). Obviously \(N_{G}(M)\) is a subgroup by \(G\). If \(U\) is a subgroup of \(G\), then \(U\leq N_{G}(U)\) holds. Apparently is \(N_{G}(M^{X} = N_{G}(M)^{X}\) for all \(X\in G\).
2. The set \(C_{G}(M) = \{g\mid g\in G, gm=mg\mbox{ for all }m\in M\}\) is called the Centralizer of \(M\) in \(G\). \(C_{G}(M)\) is also a subgroup of \(G\), and obviously \(C_{G}(M)\leq N_{G}(M)\) holds. If \(U\) is an Abelian subgroup of \(G\), then we have \(U\leq C_{G}(U)\).
3. We set \(C_{G}(G) = Z(G)\) and call this the Center of G. So the center \(Z(G)\) of \(G\) consists of those elements of \(G\), which commute with every element of \(G\).

The index \([G:N_{G}(M)]\) is the number of distinct conjugacy classes of \(M\) in \(G\).

Let \(U\) and \(B\) be subgroups of \(G\). We call \(U g B = \{ugv\mid u\in U, v\in B\}\) (with \(g\in G\)) Double Cosets of \(U\), \(B\) in \(G\).

From \(U g B\cap U h B\neq\emptyset\) it follows \(UgB = UhB\). So the double cosets yield a disjoint decomposition of \(G\). If \(G\) is finite, then

\[G=\bigcup^{n}_{i}Ug_{i}B\quad\mbox{(disjoint)}\]

and

\[|G| = \sum^{n}_{i=1}\frac{|U||B|}{|U^{g_{i}}\cap B|}.\]

Let \(G = \langle g\rangle\) be a cyclic group of order \(o(g)\). Then there is exactly one subgroup of order \(d\) for each factor \(d\) of \(o(g)\) of \(G\), namely \(\langle g^{k}\rangle\) with \(k = o(g)/d\).

The subgroup \(N\) of \(G\) is called a Normal Subgroup of \(G\) (often also called normal subgroup or invariant subgroup) if \(N^{g}\leq N\) holds for all \(g\) from \(G\). We then write \(N\trianglelefteq G\) or \(N\triangleleft G\), if \(N\) differs from \(G\). Every group \(G\) owns the trivial normal subgroup \(G\) and \(\mathbf{1}\).

If \(G\) has only the normal subgroups \(\mathbf{1}\) and \(G\), then we call \(G\) a Simple group. Obviously every group of prime order is simple.

Let \(N\) be a subgroup of \(G\). Then the following are equivalent:

1. \(N\trianglelefteq G\).
2. \(N^{g} = N\) for all \(g\in G\).
3. \(gN=Ng\) for all \(g\in G\).
4. Every left coset of \(N\) is a right coset of \(N\).

If \(N\) is a subgroup of \(G\) with \([G : N] = 2\), then \(N\triangleleft G\).

We have \((gN)(hN)=gh N\) on the set of left cosets of subgroup \(N\) of \(G\) be well-defined multiplication operator precisely when \(N\) is normal in \(G\).

When equipped with this multiplication, the cosets of \(N\) form a group.

If \(N\) is a normal subgroup of \(G\), then the cosets of \(N\) equipped with the multiplication \((gN)(hN)=ghN\) form a group called the factor group of \(G\) by \(N\). We denote it by \(G/N\). Obviously \(|G/N| = [G : N]\) holds.

1. The mapping \(\pi\) from the group \(G\) to the group \(H\) is called a Homomorphism from \(G\) to \(H\) if for any \(g_{1},g_{2}\in G\) we have \(\pi(g_{1}g_{2}) = \pi(g_{1})\pi(g_{2})\). We write \(\pi(G) = G^{\pi}=\{\pi(g)\mid g\in G\}\) and call \(\pi(G) = G^{\pi}\) the Image of \(G\) under \(\pi\).
2. The homomorphism \(\pi\) from \(G\) to \(H\) is called an Epimorphism if \(\pi\) is a surjective mapping from \(G\) to \(H\), i.e. if \(H = G^{\pi}\) holds.
3. The homomorphism \(\pi\) of \(G\) in \(H\) is called a Monomorphism, if \(\pi\) is an injective mapping from \(G\) to \(\pi(G)\).
4. A homomorphism from \(G\) to \(G\) is called an Endomorphism of \(G\).

Let \(\pi\) be a homomorphism from \(G\) to \(H\).

1. If \(E\) is the identity element of \(G\), then \(\pi(E)\) is the identity element of \(H\).
2. For each \(g\in G\) we have \(\pi(g^{-1})= \pi(g)^{-1}\).
3. \(\pi(G) = \{\pi(g)\mid g\in G\}\) is a subgroup of \(H\).
4. If \(M\) is a generator system of \(G\), then the homomorphism is \(\pi\) is uniquely determined by knowing \(\pi(m)\) for all \(m\in M\).

1. If \(R\) is a normal subgroup of \(G\), then the mapping is \(\alpha\) with \(\alpha(g)=gR\) an epimorphism from \(G\) to \(G/R\). We call \(\alpha\) the natural homomorphism from \(G\) to \(G/R\).
2. Let \(\pi\) be a homomorphism from \(G\) to \(H\). Let \(K=\{g|g\in G,\pi(g)=e\}\). We call \(K\) the kernel of \(\pi\). Then \(K\) is a normal subgroup of \(G\), and it gives a monomorphism \(\varphi\) from \(G/K\) to \(H\) with \(\alpha\varphi=\pi\). In particular, \(G/K = \pi(G)\).

If the \(N_{i}\) with \(i\in I\) are normal subgroups of \(G\), then both their product \(\langle N_{i}\mid i\in I\rangle\) and their intersection \(\bigcap_{i \in I}N_{i}\) are normal subgroups of \(G\).

Let \(N\) be a normal subgroup of \(G\).

1. Among the subgroups \(U\) of \(G\) with \(N\leq U\leq G\) and all the subgroups of \(\bar{G} = G/N\) there is a one-to-one correspondence \(\alpha\) that preserves the lattice operations intersection and product defined by \(\alpha(U)= U/N\).
2. For \(N\leq U\leq G\) we have \([G : U] = [G/N: U/N]\).
3. If \(N\leq M\trianglelefteq G\), then \(M/N\trianglelefteq G/N\) and \((G/N)/(M/N) = G/M\).

If \(N\) is a normal subgroup and \(U\) is a subgroup of \(G\), then the product \(\langle N,U\rangle\) agrees with the product \(UN\) defined in 1.2.11 agree. In particular, then \(NU = UN\).

If \(N\) is a normal subgroup of \(G\) and \(U\) is a subgroup of \(G\), then \(N\cap U\trianglelefteq U\) and \(UN/N\cong U/(U\cap N)\).

If \(M\) and \(N\) are normal subgroups of \(G\) with \(M\cap N=\mathbf{1}\), then \(mn=nm\) holds for all \(m\) from \(M\) and all \(n\) from \(N\). We then say: \(M\) and \(N\) are elementwise commutable.

If \(M\) is a subset of \(G\), then we write \(M^{G} := \langle m^{g}\mid m\in M,g\in G\rangle\). Obviously \(M^{G}\) is the smallest normal subgroup of \(G\) which contains \(M\).

An isomorphism from \(G\) to itself is called an Automorphism of \(G\). The automorphisms of \(G\) form a group \(A(G)\) with respect to the following product definition:

For \(\alpha,\beta\in A(G)\) let \(\alpha\beta\) be defined by \((\beta\alpha)(g)=\beta(\alpha(g))\).

The group \(A(G)\) is called the Automorphism Group of \(G\). If \(G\) is finite, then obviously \(A(G)\) is also finite

Every mapping \(\alpha\) of the form \(\alpha(x) = x^{g}\) with \(g\in G\) is an automorphism of \(G\), a so-called Inner Automorphism.

Inner automorphisms form a normal subgroup \(I(G)\) of \(A(G)\). We call the group \(I(G)\) the inner automorphism group of \(G\). It is isomorphic to the factor group \(G/Z(G)\) of \(G\) modulo its center \(Z(G)\).

Let \(M\) and \(N\) be normal subgroups of \(G\) with \(M\leq N\). Let \(U\) be a subgroup of the automorphism group \(A(G)\) of \(G\). For every automorphism \(\alpha\) from \(U\), suppose \(\alpha(M)=M\) and \(\alpha(N)=N\) hold. Then the mapping \(\bar{\alpha}\) with

\(\bar\alpha(nM) = \alpha(n)M\) for \(n\in N\)

is an automorphism of \(N/M\), the assignment of \(\alpha\) to \(\bar\alpha\) gives a homomorphism of \(U\) in the automorphism group \(A(N/M)\) of \(N/M\). The kernel of this homomorphism consists of the automorphisms \(\alpha\) from \(U\) of the form \(\alpha(n) = nm(n)\) with \(m(n)\in M\) for all \(n\) from \(N\). The \(m(n)\) suffice for the functional equation \(m(n_{1}n_{2}) = m(n_{1})^{n_{2}}m(n_{2})\). (In §17 we shall have to speak in detail about this functional equation.)

In the special case \(U = I(G)\) we give for the kernel of the composite Homomorphism from \(G\) (via \(I(G)\)) to \(A(N/M)\) the notation

\(C_{G}(N/M) = \{G \mid n^{g}M = nM \mbox{ for every }n\in N\}\).

Then clearly then \(G/C_{G}(N/M)\) is isomorphic to a subgroup of \(A(N/M)\). (For \(M = \mathbf{1}\), \(C_{G} (N/\mathbf{1})\) is the centralizer \(C_{G}(M)\) introduced in 2.17b.)

Let \(N\) be a normal subgroup of \(G\) and \(\alpha\) an automorphism of \(G\) with \(\alpha(n)=n\) for all \(n\in N\) and \(gN = \alpha(g)N\) for all \(g\in G\). Then \(\alpha(g) = gT(g)\), where \(T(g)\) lies in \(Z(N)\), depends only on the coset \(gN\) and the functional equation \(T(gh) = T(g)^{h}T(h)\) from Theorem 4.3 is satisfied.

All automorphisms \(\alpha\) of the given form are pairwise commutable. If \(m\) is the exponent of \(Z(N)\), then \(\alpha^{m}=1\). In particular, if \((O(\alpha), |Z(N|) = 1\), then \(\alpha = 1\).

If \(U\) is a subgroup of \(G\), let \(C_{G}(U)\) be the centralizer of \(U\) and \(N_{G}(U)\) the normalizer of \(U\) in \(G\). Then the factor group \(N_{G}(U)/C_{G}(U)\) is isomorphic to a subgroup of the automorphism group \(A(U)\) of \(U\).

Let \(G\) be a cyclic group of order \(n\). Then the automorphisms of \(G\) have the form \(\alpha(g) = g^{k}\) with \((k, n) = 1\). The automorphism group \(A(G)\) of \(G\) is isomorphic to the multiplicative group of remainders mod \(n\) that are relatively prime to \(n\). In particular, \(|A(G)| = \varphi(n)\), where \(\varphi\) is the Euler totient function from number theory.

The subgroup \(U\) of \(G\) is called a Characteristic Subgroup of \(G\) if \(\alpha(U) = U\) for every automorphism \(\alpha\) of \(G\). (Obviously every characteristic subgroup of \(G\) is a normal subgroup of \(G\).) If \(G\) has only the characteristic subgroups \(\mathbf{1}\) and \(G\), then \(G\) is called characteristically simple.

Trivial but useful is:

1. Let \(N\) be a normal subgroup of \(G\) and \(M\) a characteristic subgroup of \(N\). Then \(M\) is a normal subgroup of \(G\).
2. If \(A\leq B\leq G\), \(A\) is characteristic in Bund \(B\) is characteristic in \(G\), then \(A\) is characteristic in \(G\).

Let \(N\triangleleft G\) and let \(|N|\) and \(|G/N|\) be finite and relatively prime.

1. If \(M\leq G\) and \(|M|\) divides \(|N|\), then \(M\leq N\) holds.
2. \(N\) is the only subgroup of \(G\) with order \(|N|\).

In particular, \(N\) is a characteristic subgroup of \(G\).

1. Let \(p\) be an odd prime number and let \(G\) be the group of all integers linear maps \(x\to ax+b\) (\(a,b,x\in GF(p)\), \(a\neq0\)) over the finite field \(GF(p)\). Then every automorphism of \(G\) is an inner automorphism.
2. (G.E. WALL) Let \(G\) be the group of integral linear maps \(x\to ax + b\) over the ring of integers modulo 8 with \(a = 1 (2)\). (The order of \(G\) is 25.) Then the map is \(\alpha\), which transforms \(G\) with \(xG =ax+b\) into \(\alpha(G)\) with \(x\alpha(G) = ax + b + \frac{1}{2 }(a^{2}-1)\) is an automorphism of \(G\) which maps every element of \(G\) to a conjugate, but which is not an inner automorphism of \(G\).

Let \(\Omega\) be a finite set with \(|\Omega| = n\). We denote the elements of \(\Omega\) using the natural numbers 1, …, \(n\) (to distinguish them from group elements).

The set of all one-to-one mappings from \(\Omega\) to itself becomes a group if we define the product \(gh\) of two mappings \(g\) and \(h\) by \(i^{gh}=(i^{g})^{h}\) \((i\in \Omega)\).

We call this group the Symmetric Group \(S_{n}\) of degree \(n\) (on \(\Omega\)). Obviously \(|S_{n}|=n!\) holds. We call the elements of \(S_{n}\) permutations and often write them in the form \(G = \binom{i}{i^{g}}\). If \(\Delta\) is a subset of \(\Omega\) and \(g\in S_{n}\), we write \(\Delta^{g} = \{i^{g}\mid i\in \Delta\}\).

A permutation of shape

\[ P = \begin{pmatrix}a_{1} & a_{2} & \dots & a_{k-1} & a_{k} & a_{k+1} & \dots & a_{n}\\ a_{2} & a_{3} & \dots & a_{k} & a_{1} & a_{k+1} & \dots & a_{n}\end{pmatrix}\]

(or in short: \(P = (a_{1}, a_{2}, \dots, a_{k})\)) means a \(k\) cycle or cycle of length \(k\).

We call the 2-cycles transpositions.

Every permutation \(g\) can be written uniquely (up to ordering) as the product of disjoint cycles, which do not have a common digit.

The fixed digits \(i\) at \(g\) appear with \(i^{g} = i\) as cycles of the length 1 and are usually not written down. If the lengths of these cycles \(z_{1},\dots,z_{m}\), then the order \(o(g)\) of \(g\) is the least common multiple of \(z_{i}\). We call the \(m\) tuple \((z_{1}, \dots, z_{m})\) the Permutation type of \(g\).

Let \(g\) have the factorization into cycles \(g = (a_{1},\dots,a_{k})\dots(s_{1},\dots,s_{m})\) and let \(T = \binom{a_{i}}{b_{i}}\dots\binom{s_{j}}{t_{j}}\). Then \(g^{T} = T^{-1}gT\) has the cycle decomposition \(g^{T} = (b_{1}, \dots, b_{k})\dots(t_{1},\dots,t_{m})\).

In particular, it follows that the conjugation classes \(S_{n}\) consist of permutations of the same type.

Let \(g\) be a Permutation of \(\{1,\dots,n\}\). Write

\[ \prod_{i\lt j}(g(i)-g(j))=\varepsilon\prod_{i\lt j}(i-j) \]

with \(\varepsilon=\pm1\). We define the Sign of \(g\) to be \(\mathrm{sgn}(g)=\varepsilon\).

We call permutations \(\pi\in S_{n}\) with \(\mathrm{sgn}(\pi)=1\) Even, and those with \(\mathrm{sgn}(\pi)=-1\) Odd.

Let \(n\gt 1\). The permutations \(\pi\in S_{n}\) with \(\mathrm{sgn}(\pi)=1\) form a normal subgroup \(A_{n}\lt S_{n}\) of index 2. We call \(A_{n}\) the Alternating Group of degree \(n\).

Each subgroup of \(S_{n}\) is called a Permutation Group of degree \(n\).