Where the work of others has been used, this has been duly acknowledged in the text. NCG N is a proper normal subgroup G G/N of the factor or quotient group G on N [G:H] index H in G. The study of this topic is one of the classic areas of finite group theory.
This topic is largely motivated by Cayley's theorem, which states that every group Gi is isomorphic to a subgroup of a symmetric group. Therefore, we consider different classes of finite groups and treat each case according to the group structure. Within the work done by the authors in [20] and [36], the conditions under which the minimum degree of the faithful permutation representation of the direct product is equal to the sum of the minimum degrees of the faithful permutation representation of the direct factors are investigated.
We provide a self-contained and detailed account of the results of the investigation by the authors in [20] and [36] that will be relevant to the work in this thesis. Standalone and independent evidence of the results found by the authors of the above articles is provided.
Morphisms of groups
We only provide the proofs for the results that may be useful in the sequel, as most can be found in the relevant literature. If there is an injective homomorphism ρ:G→H, we say that G is embedded in H. If there is an on homomorphism ρ :G → H, we say that H is a homomorphic image of G.
Permutation representations
For finite groups, Cayley's Theorem gives the existence of a permutation representation, but it does not guarantee the minimality of the degree of the permutation representation. From Cayley's Theorem and Remark 2.2.1, it appears to be a plausible task to search for a permutation representation of the smallest degree for various classes of finite groups. If ρ:G→SX is a permutation representation of GonX, then there exists an actionσ of GonX corresponding to ρ. Conversely, if G acts on a nonempty set X, then there exists a permutation representation ρ:G→SX corresponding to this action.
Let ρ : G → SX be a permutation representation of G on X and identify the image of every g ∈ G with ρg on SX. Conversely, suppose we have an action σ of GonX. We show that each g∈G a permutation representation of G onX. In short: if we have a shift representation, we can build a group action and vice versa.
It is worth noting that an action corresponding to a transitive permutation representation in the sense of Theorem 2.2.2 is also transitive and vice versa. Recall that a permutation representation is finite if the permuted set is finite.
Permutation representations by acting on the cosets
The core of ρ in Theorem 2.3.1 is called the core of H in G, and is abbreviated as coreG(H). Not only do the subgroups induce a transitive permutation representation, a weak converse is also valid, that is, any transitive permutation representation is equivalent to a permutation representation induced by a special type of subgroups. Then σ is equivalent to a permutation representation by left multiplication on the left cosets of a subgroup of G, as in Theorem 2.3.1.
If we chose x=σg(x0), then we have τ(x) =gH. Now, we have σ :G→SX and by Theorem 2.3.1 we have ρ:G→SG/H, which is a transitive representation of left cosine shift with left multiplication.
The degree of a minimal faithful permutation representation . 18
If G is embedded in SX, then the embedding is equivalent to a permutation representation H={Hi}ni=1. This chapter is devoted to the study of the behavior of µ(G×H) in relation to µ(G) and µ(H). In [20, Theorem 2] it was proved that if we have a direct product of two finite groupsGandH, then the minimum degree never exceeds the sum of the minimum degrees of G and H, i.e. µ(G×H)≤µ(G) +µ(H). A necessary condition for which the inverse inequality is true is also given in [20, Theorem 2]. We plan to provide a detailed proof of this result and investigate other conditions for which the inverse inequality is true.
We prove that for any finite groups G and H, µ(G) +µ(H) is an upper bound for µ(G×H). To achieve this we need to prove the following lemma.
The additivity property of µ
Primitivity of subgroups in a representation
The last equality of the calculation arises from the fact that H < Ki for 1≤ i≤n and therefore H < Tn. So primitive subgroups of a finite groupGa simply meet irreducible elements of the subgroup lattice ofG.Let L be a finite lattice andx∈ L.Ifxis meets-irreducible thenx is a convergence of some irreducible elements of the latticeL because x = x ∧x. Suppose x is not measurement-irreducible and x = x1 ∧x2 for some x1, x2∈ L. Ifxi is not measurement-irreducible, factor it asxi =xi1∧xi2.
If xij is not meeting irreducible, then decompose it as xij =xij1∧xij2. Since L is finite, this process will terminate after a finite number of iterations. Therefore, the element itself of a finite lattice is a meeting of some meeting-irreducible elements of the lattice. So M is not strictly contained in any subgroup of the subgroups of G. Therefore, ˆM = Ø and soM 6= ˆM . Hence M is primitive.
Since the subgroup lattice of H is a chain, i.e., every subgroup of H is contained in all subgroups of H of higher order, so the H-closure of {1H} is the non-trivial subgroup of minimal order. Due to the importance of the mechanism used to prove this result, following a similar argument as in the proof of [20, Lemma 1], we provide a very detailed proof of it.
The additivity of µ for groups of coprime order
We have provided the first condition for the additivity of µ. The following appears as a converse statement of [20, Proposition 2]. Note that only {1H} can be a normal subgroup of H contained in each of the elements of H. A similar argument shows that {1G} is the only normal subgroup of G contained in each of the elements of R.
Minimal permutation representations of a direct product of
Some characterisations of finite nilpotent groups
The first characterization for finite nilpotent groups that is of interest to us is presented in the following theorem. Then the following conditions are equivalent. i) G is a non-trivial nilpotent group. ii) Every non-trivial homomorphic image of Ghas a non-trivial center. iii) G appears as a member of its central series. If Zi(G) is the second term of the upper central series of G, then Zi+1/Zi(G) = Z(G/Zi(G)) is non-trivial since G/Zi(G) is a homomorphic image of G.
The latter, together with the fact that G is a finite group, implies that not every term in the upper central row will be properly inG. Since our interest is in describing the structure of a finite nilpotent group using finitep groups, we start by first proving that finitep groups are themselves nilpotent. We know that Z(G) E G and since Z(G) 6={1G}, then the quotient group G/Z(G) is a p-group of order less than|G|. Therefore, the induction hypothesis G/ Z(G) is a nilpotent group.
We are now ready to give a characterization of nilpotent groups as constructed from their Sylow p-subgroups. The following theorem and its proof will be used extensively to prove the additivity property of µfor finite nilpotent groups in the following subsection. Now suppose G is a direct product of Sylowpi subgroups, for different primespi, say G=P1×P2× · · · ×Pn.and suppose the result is true for groups of order less than |G| = Πni=1pαii.
The additivity of µ for finite nilpotent groups
Using Theorem 3.4.14, we derive the following corollary, which deals with the additivity property µ for finite abelian groups. In the above corollary, we have µ(G×H) =µ(G) +µ(H) for finite abelian groups G and H. The finite abelian group G can be decomposed into the direct product G=G1×G2× · · · ×Gn of cyclic groups such that everyGi holds of order pαii, where every pi is a partition of |G|: this is known as the fundamental theorem of finite Abelian groups. We will derive this result from the proof of Theorem 4.1.6, where we show that if the order of a finite abelian groupG is pαii, then µ(G) =pαii.
The class G of D.Wright
The additivity of µ for the class G
Then ρ is a homomorphism and the following holds:. iii) kerρ and G/Imρ are isomorphic to the elementary abelian group of order pn. Furthermore, rank(kerρ) =rank(G/Imρ) =rank(G) =n. That is, kerρ and G/Imρ are isomorphic to the elementary abelian group of order pn. It follows by Lemma 4.1.1 (iii) that kerρ is an elementary abelian group of order p3 or p2 respectively.
There are exactly p+ 1 unique p-subgroups of order p of an elementary abelian p-group of order p2. We prove this fact in the following lemma. So the group G/P is ap-group of order pk. We consider two cases, namely the case where G/P is cyclic and the case where G/P is non-cyclic. That is, kerρ consists precisely of the elements in M of order p. Now, if β ∈Aut(M), then o(β(x)) =p, since β preserves the order of elements.
Now, if|kerρ|=p2, thenkerρ is isomorphic to either Cp2orCp×Cp. However, kerρ has no element of order p2. However, we will also use Theorem 4.1.4 and Theorem 4.1.5 for the remainder of this dissertation. ii) If P is an ap-group with at most one subgroup of order p, then either P is cyclic orp= 2 and P is isomorphic to Q2n, n≥3. Since Z(Q2n) is the unique subgroup of order 2, and by Lagrange's theorem, we have that all non-trivial subgroups of Q2n contain the normal subgroup Z(Q2n).
We will prove this directly, i.e. we show that n = 7 is the smallest integer such that Q is embedded in Sn. To achieve this, we construct G≤S7 such that Q∼=G. It follows that Q and G have the same structure: essentially Q∼=G under the map a7 → α and b7 → β, that is, we map the generators of Q to the generators G. Since P N/N is the unique subgroup is of M/N of order |PN /N|, it follows that ϕ|P N/N, the restriction of the automorphism ϕ from M/N to P N/N, is an injection of P N/N into a subgroup of order | P N/N|inM/N .
An order element greater than pαr−1 inG/N will be an image of one of these. In Theorem 5.4.8 and Theorem 5.4.9 we have given three exceptional groups of order p5 with distinctive quotients isomorphic to G16, G25 and G26. In each of the exceptional groups of order p5 from Theorem 5.4.8 and Theorem 5.4.9, G is an extension of its distinct quotient by the distinct subgroup N =hwi ≤Z(G). That is, G is a central extension of G16, G25 or G26, with N = hwi.
If G is an exceptional group of order p5, for odd p, then G is a distinct quotient extension G/N of order p4 with a central subgroup N of order p, and G/N is isomorphic to G16, G25, G26, or G27. By Theorem 5.4.7, we know that for an odd prime number p there are no exceptional groups p of order lower than p5. For an odd prime, by Theorem 5.5.1, we know that if G is an exceptional group of order p5, then G is a central extension of a distinct G/N of order p4 with some subgroup N ≤Z(G) of order p, where G/N is isomorphic.
To answer this question, the isomorphism classes of p-groups with an order higher than p5 must of course be considered.
