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تسجيل مستخدم جديدSome characterizations of ideal variants of Hurewicz type covering properties
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In this paper, we continue to investigate topological properties of $mathcal{I}H$ and its two st
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Which Isbell--Mrowka spaces ($Psi$-spaces) satisfy the star version of Mengers and Hurewiczs covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $Psi$-space that
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Let $G$ be a finite group and let $pi(G)={p_1, p_2, ldots, p_k}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${rm GK}(G)$, is defined as follows: its vertex set is $pi(G)$ and two dif
ferent vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_ip_j$. The degree of a vertex $p_i$ in ${rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)=left(d_G(p_1), d_G(p_2), ldots, d_G(p_k)right)$ is said to be the degree pattern of $G$. Moreover, if $omega subseteq pi(G)$ is the vertex set of a connected component of ${rm GK}(G)$, then the largest $omega$-number which divides $|G|$, is said to be an order component of ${rm GK}(G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$.
Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even order, we hav
e |G| is less than the number of conjugacy classes of the Fitting subgroup times the order of the centralizer to the fourth power of any involution in G. This result does require the classification of the finite simple groups. The groups SL(2,q) with q even shows that the exponent 4 cannot be replaced by any exponent less than 3. We do not know at present whether the exponent 4 can be improved in general, though we note that the exponent 3 suffices for almost simple groups G. We are however able to prove that every finite group $G$ of even order contains an involution u such that [G:F(G)] is less than the cube of the order of the centralizer of u. There is a dichotomy in the proof of this fact, as it reduces to proving two residual cases: one in which G is almost simple (where the classification of the finite simple groups is needed) and one when G has a Sylow 2-subgroup of order 2. For the latter result, the classification of finite simple groups is not needed (though the Feit-Thompson odd order theorem is). We also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups, and some other results on the existence of non-central elements (not necessarily involutions) with large centralizers in general finite groups. We also show (without the classification of finite simple groups) that if t is an involution in G and p is a prime divisor of [G:F(G)], then p is at most 1 plus the order of the centralizer of t (and this is best possible).
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