The number of automorphisms of a monolithic finite group. (English) Zbl 1193.20026
A nontrivial finite group is called monolithic if it has a unique minimal normal subgroup (being a monolithic \(p\)-group is equivalent to having a cyclic center).
The author is interested in calculating the order of \(\operatorname{Aut}(G)\) when \(G\) is monolithic. To do this, he detects, for each prime-power \(q>1\) with \((q,|G|)=1\) the minimal degree \(m=\min\deg(G,q)\) among all the faithful \(F\)-representations of \(G\) over the field \(F\) with \(q\) elements. Then he introduces the notion of a good monolithic triple (here character theory comes into play) and derives a formula for \(|\operatorname{Aut}(G)|\) depending on the number of some special characters and also the orders of normalizers of special subgroups (isomorphic to \(G\)) of the general linear group \(\text{GL}(m,q)\).
This approach seems strange at first glance. After all, it is well known that the order of \(\operatorname{Aut}(G)\) is the index of a copy of \(G\) in its normalizer with respect to the symmetric group \(\text{Symm}(G)\) via the holomorph construction. This approach is cheaper from the computational point of view (for it is well known that general linear groups in dimension \(n\) are very larger than symmetric groups on \(n\) letters and besides, the author’s approach is restricted to monolithic groups).
However, this original approach leads the author to very interesting applications and also very interesting theoretical consequences relating to normalizers of special subgroups in general linear groups over finite fields.
The paper is very well written, almost self-contained, and a joy to read.
The author is interested in calculating the order of \(\operatorname{Aut}(G)\) when \(G\) is monolithic. To do this, he detects, for each prime-power \(q>1\) with \((q,|G|)=1\) the minimal degree \(m=\min\deg(G,q)\) among all the faithful \(F\)-representations of \(G\) over the field \(F\) with \(q\) elements. Then he introduces the notion of a good monolithic triple (here character theory comes into play) and derives a formula for \(|\operatorname{Aut}(G)|\) depending on the number of some special characters and also the orders of normalizers of special subgroups (isomorphic to \(G\)) of the general linear group \(\text{GL}(m,q)\).
This approach seems strange at first glance. After all, it is well known that the order of \(\operatorname{Aut}(G)\) is the index of a copy of \(G\) in its normalizer with respect to the symmetric group \(\text{Symm}(G)\) via the holomorph construction. This approach is cheaper from the computational point of view (for it is well known that general linear groups in dimension \(n\) are very larger than symmetric groups on \(n\) letters and besides, the author’s approach is restricted to monolithic groups).
However, this original approach leads the author to very interesting applications and also very interesting theoretical consequences relating to normalizers of special subgroups in general linear groups over finite fields.
The paper is very well written, almost self-contained, and a joy to read.
Reviewer: Marian Deaconescu (Safat)
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20C15 | Ordinary representations and characters |
| 20G40 | Linear algebraic groups over finite fields |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20F40 | Associated Lie structures for groups |
Keywords:
finite monolithic groups; finite \(p\)-groups; automorphism groups; orders of groups of automorphisms; character theory; finite general linear groupsReferences:
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