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Register a new userOn the (2,3)-generation of the finite symplectic groups
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This paper is a new important step towards the complete classification of the finite simple groups which are $(2, 3)$-generated. In fact, we prove that the symplectic groups $Sp_{2n}(q)$ are $(2,3)$-generated for all $ngeq 4$. Because of the existing literature, this result implies that the groups $PSp_{2n}(q)$ are $(2,3)$-generated for all $ngeq 2$, with the exception of $PSp_4(2^f)$ and $PSp_4(3^f)$.
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In this paper we prove that the unitary groups $SU_n(q^2)$ are $(2,3)$-generated for any prime power $q$ and any integer $ngeq 8$. By previous results this implies that, if $ngeq 3$, the groups $SU_n(q^2)$ and $PSU_n(q^2)$ are $(2,3)$-generated, except when $(n,q)in{(3,2),(3,3),(3,5),(4,2), (4,3),(5,2)}$.
In this paper we give explicit (2,3)-generators of the unitary groups SU_6(q^ 2), for all q. They fit into a uniform sequence of likely (2,3)-generators for all n>= 6.
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We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of $U$, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).
In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.
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