Clara Franchi

• PhD
• Professor (Associate) at Università Cattolica del Sacro Cuore

We use Majorana representations to study the subalgebras of the Griess algebra that have shape $(2B,3A,5A)$ and whose associated Miyamoto groups are isomorphic to $A_n$. We prove that these subalgebras exist only if $n\in \{5,6,8\}$. The case $n=5$ was already treated by Ivanov, Seress, McInroy, and Shpectorov. In case $n=6$ we prove that these alg...

Published version of the previous preprint

Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(\alpha, \beta)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this...

In [15], Yabe gives an almost complete classification of primitive symmetric 2-generated axial algebras of Monster type, leaving open the case of algebras in characteristic 5 and axial dimension larger than 5. In this note, we complete the classification of these algebras, by constructing a new infinite-dimensional primitive 2-generated symmetric a...

Free access Abstract and full article downloadable at the following link: https://rdcu.be/cygWp

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type $$(\alpha ,\beta )$$ ( α , β ) , over a field of characteristic other than 2, has dimension at most 8 if $$\alpha \notin \{2\beta ,4\beta \}$$ α ∉ { 2 β , 4 β } . In this n...

In this paper we provide the basic setup for a project, initiated by Felix Rehren, aiming at classifying all $2$-generated primitive axial agebras of Monster type $(\alpha, \beta)$. We first revise Rehren's construction of an initial object in the cathegory of primitive $n$-generated axial algebras giving a formal one, filling some gaps and, though...

In this paper we prove that $2$-generated primitive axial algebras of Monster type $(2\beta, \beta)$ over a ring $R$ in which $2$ and $\beta$ are invertible can be generated as $R$-module by $8$ vectors. We then completely classify $2$-generated primitive axial algebras of Monster type $(2\beta, \beta)$ over any field of characteristic other than $...

Recently Takahiro Yabe gave an almost complete classification of primitive symmetric $2$-generated axial algebras of Monster type. In this note, we construct a new infinite-dimensional primitive $2$-generated symmetric axial algebra of Monster type $(2, \frac{1}{2})$ over a field of characteristic $5$, and use this algebra to complete the last case...

Rehren proved that a primitive 2-generated axial algebra of Monster type $(\alpha,\beta)$ has dimension at most eight if $\alpha\notin\{2\beta,4\beta\}$. In this note we construct an infinite-dimensional 2-generated primitive axial algebra of Monster type $(2,\frac{1}{2})$ over an arbitrary field $F$ with $char(F)\neq 2,3$. This shows that the seco...

We determine all saturated fusion systems F on a Sylow 3-subgroup of the sporadic McLaughlin group that do not contain any non-trivial normal 3-subgroup and show that they are all realizable.

Majorana representations have been introduced by Ivanov in [15] in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer invo-lutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A 12 (by [7], the la...

In this paper we give a brief overview of the theory of Majorana representations of finite groups.

As a consequence of a general result about finite group actions on 3-manifolds, we show that a hyperbolic 3-manifold can be the cyclic branched cover of at most fifteen inequivalent knots in S³ (in fact, a main motivation of the present paper is to establish the existence of such a universal bound). A similar, though weaker, result holds for arbitr...

In this paper we present a general method for computing the irreducible components of the permutation modules of the symmetric groups over a field F of characteristic 0. We apply this machinery to determine the decomposition into irreducible submodules of the F [Sn]-permutation module on the right cosets of the normaliser in Sn of the subgroup gene...

Let G be a finite group, W be a \({\mathbb {R}}[G]\)-module equipped with a G-invariant positive definite bilinear form \((\,,\,)_W\), and X a finite generating set of W such that X is transitively permuted by G. We show a new method for computing the dimensions of the irreducible constituents of W. Further, we apply this method to Majorana represe...

We show that all 2A-Majorana representations of the Harada-Norton group F5 have the same shape. If R is such a representation, we determine, using the theory of association schemes, the dimension and the irreducible constituents of the linear span U of the Majorana axes. Finally, we prove that, if R is based on the (unique) embedding of F5 in the M...

We show that a finite group of orientation preserving diffeomorphisms of a $3$-manifold which is not the $3$-sphere can contain at most fifteen conjugacy classes of cyclic subgroups acting with non-empty and connected fixed-point sets and having the $3$-sphere as space of orbits. A straightforward corollary of this fact is that a hyperbolic $3$-man...

For a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′)≤μ(G).

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if \[ \Pr(\min \pi(X)=\pi (x))=\frac{1}{|X|} \] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property th...

We show that Lyons' sporadic simple group is the unique finite simple group G that is simultaneously of weak characteristic r-type and characteristic p-type for odd primes r and p with r<p and mr,p(G)⩾2.

A permutation group G of finite degree n is a sharp irredundant group of type {k}, k a positive integer, if each non-identity element of G fixes exactly k points, |G|=n−k and G has no global fixed point and no regular orbit. In this note we give a method to construct all faithful representations of finite abelian groups as sharp irredundant permuta...

A permutation groupG of finite degreed is called a sharp permutation group of type {k},k a non-negative integer, if every non-identity element ofG hask fixed points and |G|=d−k. We characterize sharp non-abelianp-groups of type {k} for allk.

Let Ω be a finite set of cardinality n with a linear order on it and let k be a positive integer. Let F be a set of permutations on Ω and let PrF be an arbitrary distribution of probability on F. The set F is said to be biased k-restricted min-wise independent if for every subset X of Ω such that |X|≤k, and every x∈X, when π is chosen at random in...

We study soluble groups in which every subgroup lying between a characteristic subgroup and its derived subgroup is normal.

A permutation group G is said to be a group of finite type { k }, k a positive integer, if each non-identity element of G has exactly k fixed points. We show that a group G can be faithfully represented as an irredundant permutation group of finite type if and only if G has a non-trivial normal partition such that each component has finite bounded...

We consider groups Γ of automorphisms of a groupG acting by means of power automorphisms on the factors of a normal series inG with lengthm. We show that [G, Γ] is nilpotent with class at mostm and that this bound is best possible. Moreover, such a Γ is parasoluble with paraheight at most 1/2m(m+3)+1, provided Γ′ is periodic. We give best possible...

In a group G, um (G) denotes the subgroup of the elements which normalize every subnormal subgroup of G with defect at most m. The m-Wielandt series of G is then defined in a natural way. G is said to have finite m-Wielandt length if it coincides with a term of its m-Wielandt series. We investigate the structure of infinite groups with finite m-Wie...

For each m≥1 , u_{m}(G) is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most m in G . An ascending chain of subgroups u_{m,i}(G) is defined by setting u_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)) . If u_{m,n}(G)=G , for some integer n , the m - Wielandt length of G is the minimal of such n . In [ 3...