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A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied to study

For positive integers k and n, the shuffle group $G_{k,kn}$ is generated by the $k!$ permutations of a deck of $kn$ cards performed by cutting the deck into k piles with n cards in each pile, and then perfectly interleaving these cards following a certain permutation of the k piles. For $k=2$ , the shuffle group $G_{2,2n}$ was determined by Diaconis, Graham and Kantor in 1983. The Shuffle Group Conjecture states that, for general k, the shuffle group $G_{k,kn}$ contains $\\mathrm {A}_{kn}$ whenever $k\\notin \\{2,4\\}$ and n is not a power of k. In particular, the conjecture in the case $k=3$ was posed by Medvedoff and Morrison in 1987. The only values of k for which the Shuffle Group Conjecture has been confirmed so far are powers of $2$ , due to recent work of Amarra, Morgan and Praeger based on Classification of Finite Simple Groups. In this paper, we confirm the Shuffle Group Conjecture for all cases using results on $2$ -transitive groups and elements of large fixed point ratio in primitive groups.

The classical problem of whether 𝑚th-powers with or without zero in a finite field 𝔽𝑞 form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this paper new necessary and sufficient conditions are established including those via a system of polynomial equations on Gauss sums. The author thereby solves the problem for even 𝑞 which is neglected in the literature, and extends the nonexistence list for even 𝑚 up to 22. Moreover, conjectures toward the complete classification are posed.

In a graph Γ with vertex set V , a subset C of V is called an ( a ,  b )-regular set if every vertex in C has exactly a neighbors in C and every vertex in V \\ C has exactly b neighbors in C , where a and b are nonnegative integers. In the literature, (0, 1)-regular sets are known as perfect codes and (1, 1)-regular sets are known as total perfect codes. In this paper, we prove that, for any finite group G , if a non-trivial normal subgroup H of G is a perfect code in some Cayley graph of G , then for any pair of integers a and b with 0 ⩽ a ⩽ | H | - 1 and 0 ⩽ b ⩽ | H | such that gcd ( 2 , | H | - 1 ) divides a , H is also an ( a ,  b )-regular set in some Cayley graph of G depending on ( a ,  b ). A similar result involving total perfect codes is also proved in the paper.

For groups \\(G\\) that can be generated by an involution and an element of odd prime order, this paper gives a sufficient condition for a certain Cayley graph of \\(G\\) to be a graphical regular representation (GRR), that is, for the Cayley graph to have full automorphism group isomorphic to \\(G\\). This condition enables one to show the existence of GRRs of prescribed valency for a large class of groups, and in this paper, \\(k\\)-valent GRRs of finite nonabelian simple groups with \\(k\\geq5\\) are considered.

This paper studies the existence of finite non-Desarguesian flag-transitive projective plane, giving necessary conditions in terms of polynomial equations over finite fields of characteristic \\(3\\). This sheds light on the longstanding conjecture that every finite flag-transitive projective plane is Desarguesian.

For positive integers \\(k\\) and \\(n\\), the shuffle group \\(G_{k,kn}\\) is generated by the \\(k!\\) permutations of a deck of \\(kn\\) cards performed by cutting the deck into \\(k\\) piles with \\(n\\) cards in each pile, and then perfectly interleaving these cards following a certain permutation of the \\(k\\) piles. For \\(k=2\\), the shuffle group \\(G_{2,2n}\\) was determined by Diaconis, Graham and Kantor in 1983. The Shuffle Group Conjecture states that, for general \\(k\\), the shuffle group \\(G_{k,kn}\\) contains \\(\\mathrm{A}_{kn}\\) whenever \\(k\\notin\\{2,4\\}\\) and \\(n\\) is not a power of \\(k\\). In particular, the conjecture in the case \\(k=3\\) was posed by Medvedoff and Morrison in 1987. The only values of \\(k\\) for which the Shuffle Group Conjecture has been confirmed so far are powers of \\(2\\), due to recent work of Amarra, Morgan and Praeger based on Classification of Finite Simple Groups. In this paper, we confirm the Shuffle Group Conjecture for all cases using results on \\(2\\)-transitive groups and elements of large fixed point ratio in primitive groups.

We study the normal Cayley graphs \\(\\mathrm{Cay}(S_n, C(n,I))\\) on the symmetric group \\(S_n\\), where \\(I\\subseteq \\{2,3,\\ldots,n\\}\\) and \\(C(n,I)\\) is the set of all cycles in \\(S_n\\) with length in \\(I\\). We prove that the strictly second largest eigenvalue of \\(\\mathrm{Cay}(S_n,C(n,I))\\) can only be achieved by at most four irreducible representations of \\(S_n\\), and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when \\(I\\) contains neither \\(n-1\\) nor \\(n\\) we know exactly when \\(\\mathrm{Cay}(S_n, C(n,I))\\) has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of \\(S_n\\), and we obtain that \\(\\mathrm{Cay}(S_n, C(n,I))\\) does not have the Aldous property whenever \\(n \\in I\\). As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of \\(\\mathrm{Cay}(S_n, C(n,\\{k\\}))\\) where \\(2 \\le k \\le n-2\\).

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order \\(2^s\\) for every \\(s\\geqslant3\\) has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer \\(\\mathrm{D}_8^2\\times\\mathrm{C}_2^m\\) for each integer \\(m\\geqslant1\\), where \\(\\mathrm{D}_8^2\\) is the direct product of two copies of the dihedral group of order \\(8\\) and \\(\\mathrm{C}_2^m\\) is the direct product of \\(m\\) copies of the cyclic group of order \\(2\\). The graphs constructed have surprisingly many significant properties in various contexts.

Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the \"normal\" case of Aldous' spectral gap conjecture.