The aim of this thesis is to study the relationship between the derivative length of the group of units in a group algebra of a finite group over a field with finite characteristics and the commutativity of the group. In particular, we studied group algebras whose unit groups have a derivative length of up to four, and proved their commutativity when the characteristic of the field was p≥17. Since the derivative length of the group of units is related to the strong Lie derivative length of the group algebra, we also studied group algebras with a strong Lie derivative length of up to four.
The study of the ring-theoretic properties of group algebras is now at the most advanced level. Kurdish for the investigation of derivative length, the nilpotency class and the Engel length of the group of units. Our first problem in Chapter 2 assumes that the unit group U of the group algebra satisfies the relation (U(3), U0) = 1, that is, U belongs to the class of groups of derived length four.
Geometrically, Dm can be defined as the group of isometries of the m-sided regular polygon. The strong Lie-derived series are therefore an important tool for investigating the derived length of U. Let K be a field with finite characteristic p, and Op(G) be a maximum normal p-subgroup of the finite group G.
Then the group U(KG) is solvable if and only if one of the following statements holds:. ii) G/Op(G) is abelian and K is a field of characteristic p. Op(G) also stands for the maximal normal p-subgroup of G, and ∆(G) denotes the complement ideal of the group algebra KG. We also prove commutativity of the group G of odd order pm [(p, m) = 1] when the derivative length of the unitary group U is small compared to the characteristic p of the field K.
Key Steps in the Proof of Theorem 3.1
Let K be a field of characteristic p and G be any group of odd order pm, where m is co-prime to p. We denote the Frattini subgroup of a group G by Φ(G), which is the intersection of all maximal subgroups of G. We can easily extend Theorem 3.2 to any group G of odd order pnm with (p, m) = 1 provided that the quotient of the itsp-Sylow subgroup of the Frattini subgroup is cyclic.
Let K be a field with characteristic p and G a group of odd order pnm where m is co-prime with p. If the quotientP/Φ(P) is cyclic and the derivative length of U is strictly less than dlog2(2p)e, then G must be abelian. i). First we will show that G can be written as a semi-direct product of P and H, where P is a p-group and H is an abelian p0-group. ii).
Our goal is then to express G as a direct product of P and H. If not, then there will be an element h in H that induces a non-identity p0-automorphism on P. iii). When P is elementary abelian, we will show that the non-identity p0-automorphism on P induced by h can be used to construct a non-trivial element in U(4). iv). Next, exploiting the Frattini subgroup, we will reduce the argument for general P to the case where P is elementary abelian.
Since the Frattini subgroup Φ(P) is a characteristic subgroup of P, hΦ(P) = Φ(P) and therefore h induces an automorphism on P/Φ(P). Now we have already proven thath induces identity automorphism on the elementary abelian group P/Φ(P). There is a well-known result from Burnside (Theorem 1.17) which states that if ψ is a p0-automorphism of a p-group P, which induces the identity on P/Φ(P), then ψ is the identity automorphism on P. Therefore, we conclude In our case, using the above result, we find that h also induces the identity automorphism on P. v) The fact that P will also be abelian now follows easily from the above subcase involving p-groups.
Proof of Theorem 3.1
In this subsection we want to prove that G will be abelian, i.e. G will be a direct product of the above p-group and the abelian p0 group. We first prove the following lemma, which is the most crucial ingredient in establishing Theorem 3.1. Clearly, it will be enough to show non-triviality modulo an appropriate power of the Jacobson radical J.
We now proceed to form commutators of suitable elements in U and obtain elements in U0, U(2), U(3) and finally in U(4).
When the Derived Length of U is Smaller than dlog 2 (2p)e
Proof of Theorem 3.2
Then hC is a non-trivial element of order that divides l in G/C and the corresponding power of hC gives an element of prime order q in G/C. According to Result 3.5, we can conclude that the derived length U(KG) must be at least dlog2(2p)e. However, it contradicts our assumption that the derived length U(KG) is less than dlog2(2p)e.
Proof of theorem 3.3
A classic result by Passi, Passman and Sehgal ([21]) states that whenp >2 andG is a non-abelian group, the group algebra KG is Lie-solvable if and only if KG is strongly Lie-solvable. Unfortunately, no general formula is known to calculate the Lie-derived length dlL(KG), while it is possible to give a more accurate estimate of dlL(KG). Very simple calculations allow one to conclude that this lower bound also applies to the strong Lie derivative length of a group algebra.
Along this path, Balogh and Juh´asz characterized group algebras whose strong Lie derivative length is exactly dlog2(p+1)e. Then dlL(KG) = dlog2(p+ 1)e if and only if one of the following conditions holds:. i) p= 2 and G0 is a central elementary abelian subgroup of order 4;. For p > 13 there are no strongly Lie solvable group algebras with a strongly Lie derivative of length 4. by Shalev bounds or a simple observation in [15]).
Now, in this chapter, we have given an independent proof of the characterization of KG when dlL(KG) ≤ 4 which does not involve Lie-derived sequences of KG for the cases p≥ 11 . For the cases p we have given sufficient conditions for KG to be strongly Lie solvable with strongly Lie derived length 4. Our main results in this chapter are as follows, where theorems 4.3 and 4.4(I),(II) already existing results is as mentioned above, but has been independently proven.
Let K be a field with characteristic 3≤p≤11 and let G be a group and any of the following holds. In this section we discuss a few important known results which provide useful tools to prove our theorem. We first state the complete set of necessary and sufficient conditions for KG to be solvable in Lie, given by Passi, Passman and Sehgal ([21]).
Let KG be the group algebra of G over the field K with characteristic p≥0. i) KG is Lie nilpotent if and only if G is p-abelian and nilpotent;. The next two results were proven by Sahai ([29], Remark 2.1 and Lemma 2.2 and the section before).
Main Results
Thus, we can extend this result to the entire KG group algebra and our desired result is proved. Again in the same way, we can show that the following n-pairs of terms for i= 1,. Thus, we can extend this element to the entire group algebra KG and our desired result is proved.
Since strong Lie solubility implies Lie solubility, we therefore have by Theorem 4.6 that G0 is a finite p-group, since char K 6= 2. The question that remains for this problem is therefore what will be the necessary conditions for a strong Lie soluble group. algebra to have a strong Lie derivative of length 4 over fields of characteristic 3 and 5. Based on our work in the thesis, a number of problems can be explored in the future.
IWhat will be the structure of KGover fields with characteristic p <17 with units of derived length four. Since our work contains ideas for generalizing the problem of characterizing group algebras with units of derivative length, our ideas can be used to answer the following very important questions. I What will be the smallest derivative length of U for group algebras for any given finite non-abelian group.
Finally, I complete the above problem by investigating whether the result can also be extended to any infinite set. Lie properties of the algebra group and the nilpower class of the unit group.
