Character degrees, class sizes, and normal subgroups of a certain class of \(p\)-groups. (English) Zbl 0932.20015
A particular class of finite \(p\)-groups \(P_n(q,e)\) and their extension groups is investigated in great detail. These \(p\)-groups have appeared at the first time in work by I. M. Isaacs [Can. J. Math. 41, No. 1, 68-82 (1989; Zbl 0686.20002)].
The group \(P_n(q,e)\) is constructed as a certain subgroup of the group of units in a truncated skew-polynomial ring \(R\) over the finite field \(\text{GF}(q^e)\). The group of units in \(R\) also contains a cyclic subgroup \(C\) of order \(c=(q^e-1)/(q-1)\) that normalizes \(P_n(q,e)\). The action of \(C\) on \(P_n(q,e)\) is a Frobenius action, and provides additional information on the structure of \(P_n(q,e)\) which plays a crucial role in the later determination of the set of character degrees of \(P_n(q,e)\).
The three defining parameters \(q\), \(n\), \(e\) have to satisfy the following two conditions: (a) \(e,n\in\mathbb{N}\) and \(n>1\), (b) \(q\) is a power of a prime \(p\), and \((e,(q-1)n!)=1\). The groups \(P_n(q,e)\) have a number of remarkable properties, some of them are listed below.
(i) \(|P_n(q,e)|=q^{en}\) and the class of \(P_n(q,e)\) is \(n\).
(ii) The upper and lower central series of \(P_n(q,e)\) coincide.
(iii) Each of the \(n\) central factors of \(P_n(q,e)\) is naturally isomorphic to the additive group of \(\text{GF}(q^e)\).
(iv) The derived length of \(P_n(q,e)\) is \(\lceil\log_2(n+1)\rceil\), which is precisely as large as possible, given the class.
(v) The class number of \(P_n(q,e)\) is \(1+(q^n-1)(q^e-1)/(q-1)\).
(vi) Conjugacy class sizes of \(P_n(q, e)\) are \(\{(q^{e-1})^i\mid i= 0,1,\dots,n-1\}\).
(vii) Irreducible character degrees of \(P_n(q,e)\) are \(\{(q^{(e-1)/2})^i\mid i=0,1,\dots,n-1\}\).
The solvable extension group \(P_nCG\) of order \(q^{en}ce\), where \(G\) is the Galois group of the field extension \(\text{GF}(q^e)/\text{GF}(q)\), is investigated. If \(e\) is a prime, then the set of character degrees of \(P_nCG\) is \(\{1,e\}\cup\text{cd}(P_n)\). The nilpotent length of that group is 3, the derived length is \(\lceil\log_2(n+ 1)\rceil+2\) and \(|\text{cd}(P_nCG)|=n+2\).
Next, we have \(\text{dl}(P_1CG)|=3=|\text{cd}(P_1CG)|\) and \(\text{dl}(P_2CG)|=4=|\text{cd}(P_2CG)|\). The last inequality meets rarely. For \(n=4\), the groups \(P_4CG\) provide the first known examples of odd order groups (this is the case if and only if the parameters \(e\) and \(q\) are both odd) satisfying \(\text{dl}(P_4CG)=5=|\text{cd}(P_4CG)|-1\). In the case when \(q= p\), the full set of normal subgroups of \(G\) is determined.
There are very few papers investigating in that great detail a large class of \(p\)-groups as this one.
The group \(P_n(q,e)\) is constructed as a certain subgroup of the group of units in a truncated skew-polynomial ring \(R\) over the finite field \(\text{GF}(q^e)\). The group of units in \(R\) also contains a cyclic subgroup \(C\) of order \(c=(q^e-1)/(q-1)\) that normalizes \(P_n(q,e)\). The action of \(C\) on \(P_n(q,e)\) is a Frobenius action, and provides additional information on the structure of \(P_n(q,e)\) which plays a crucial role in the later determination of the set of character degrees of \(P_n(q,e)\).
The three defining parameters \(q\), \(n\), \(e\) have to satisfy the following two conditions: (a) \(e,n\in\mathbb{N}\) and \(n>1\), (b) \(q\) is a power of a prime \(p\), and \((e,(q-1)n!)=1\). The groups \(P_n(q,e)\) have a number of remarkable properties, some of them are listed below.
(i) \(|P_n(q,e)|=q^{en}\) and the class of \(P_n(q,e)\) is \(n\).
(ii) The upper and lower central series of \(P_n(q,e)\) coincide.
(iii) Each of the \(n\) central factors of \(P_n(q,e)\) is naturally isomorphic to the additive group of \(\text{GF}(q^e)\).
(iv) The derived length of \(P_n(q,e)\) is \(\lceil\log_2(n+1)\rceil\), which is precisely as large as possible, given the class.
(v) The class number of \(P_n(q,e)\) is \(1+(q^n-1)(q^e-1)/(q-1)\).
(vi) Conjugacy class sizes of \(P_n(q, e)\) are \(\{(q^{e-1})^i\mid i= 0,1,\dots,n-1\}\).
(vii) Irreducible character degrees of \(P_n(q,e)\) are \(\{(q^{(e-1)/2})^i\mid i=0,1,\dots,n-1\}\).
The solvable extension group \(P_nCG\) of order \(q^{en}ce\), where \(G\) is the Galois group of the field extension \(\text{GF}(q^e)/\text{GF}(q)\), is investigated. If \(e\) is a prime, then the set of character degrees of \(P_nCG\) is \(\{1,e\}\cup\text{cd}(P_n)\). The nilpotent length of that group is 3, the derived length is \(\lceil\log_2(n+ 1)\rceil+2\) and \(|\text{cd}(P_nCG)|=n+2\).
Next, we have \(\text{dl}(P_1CG)|=3=|\text{cd}(P_1CG)|\) and \(\text{dl}(P_2CG)|=4=|\text{cd}(P_2CG)|\). The last inequality meets rarely. For \(n=4\), the groups \(P_4CG\) provide the first known examples of odd order groups (this is the case if and only if the parameters \(e\) and \(q\) are both odd) satisfying \(\text{dl}(P_4CG)=5=|\text{cd}(P_4CG)|-1\). In the case when \(q= p\), the full set of normal subgroups of \(G\) is determined.
There are very few papers investigating in that great detail a large class of \(p\)-groups as this one.
Reviewer: Yakolev Berkovich (Afula)
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20C15 | Ordinary representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D30 | Series and lattices of subgroups |
Keywords:
finite \(p\)-groups; groups of units; skew-polynomial rings; character degrees; lower central series; derived lengths; class numbers; odd order groupsCitations:
Zbl 0686.20002References:
| [1] | Isaacs, I. M.: Coprime group actions fixing all nonlinear irreducible characters. Canad. J. Math. 41, 68-82 (1989) · Zbl 0686.20002 |
| [2] | Isaacs, I. M.: Character theory of finite groups. (1994) · Zbl 0849.20004 |
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