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We study the nature of the finite-temperature chiral transition in QCD with N f light quarks in the adjoint representation (aQCD). Renormalization-group arguments show that the transition can be continuous if a stable fixed point exists in the renormalizationgroup flow of the corresponding three-dimensional Φ 4 theory with a complex 2N f × 2N f symmetric matrix field and symmetry-breaking pattern SU(2N f ) → SO(2N f ). This issue is investigated by exploiting two three-dimensional perturbative approaches, the massless minimal-subtraction scheme without expansion and a massive scheme in which correlation functions are renormalized at zero momentum. We compute the renormalization-group functions in the two schemes to five and six loops respectively, and determine their largeorder behavior. The analyses of the series show the presence of a stable three-dimensional fixed point characterized by the symmetry-breaking pattern SU(4) → SO(4). This fixed point does not appear in an -expansion analysis and therefore does not exist close to four dimensions. The finite-temperature chiral transition in two-flavor aQCD can therefore be continuous; in this case its critical behavior is determined by this new SU(4)/SO(4) universality class. One-flavor aQCD may show a more complex phase diagram with two phase transitions. One of them, if continuous, should belong to the O(3) vector universality class.

A complete description is given of finite minimal non-Z-groups, where Z is one of four grouptheoretical properties.Let Z be some group-theoretical property. A group is called a minimal non-Z-group if all its proper subgroups have the property Z, while the group itself does not. The importance of studying finite minimal non-Z-groups,for concrete r.,stems from the fact that every finite group not possessing the property Z contains as a subgroup a minimal non-Z-group, provided that the identity group has the property Z. Numerous results have been obtained in this direction, already traditional in the theory of groups. We note that ff Z 1 and Z 2 are two properties such that every r.l-grou p is also a Z2-group, then a minimal non-r.l-grou p is either a minimal non-Z2-grou p or has the property Z 2.In this note only finite groups are considered. The aim of the note is to study the structure of minimal non-Z-groups for the following properties of Z: The following property is denoted by C Z: The centralizer of every nonidentity element possesses the property r~ If CZ coincides with Z, then the center of the minimal non-Z-group is, clearly, nontrivial. The properties considered in this paper satisfy the condition CZ = r, which facilitates considerably the investigation. It turned out that the groups studied are very near to the groups of Shmidt. By a group of Shmidt is meant, as usual, a nonnilpotent group, in which all proper subgroups are ntlpotent.

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Recently there have been published a number of papers concerning the influence of the intersections of the Sylow 2-subgroups of a finite group on the structure of the group itself. The first was the paper of M. Suzuki [i] on 7"7--groups, in which he described the finite groups with trivial intersections of Sylow 2-subgroups. Most of the results in this direction are connected with restrictions on the internal structure of the intersections: cyclicity, commutativity, small rank, and so on (V. D. Mazurov, S. A. Syskin, G. Stroth, M. Herzog, and others). Gomi [2] investigated the case where the intersection of any two distinct Sylow 2-subgroups P and ~ is contained in the center of some Sylow 2-subgroup.It turns out, in this situation, that Pn~ is contained in the intersection of the centers of P and ~ .Our note is devoted to a more general problem, formulated by Mazurov and Syskin in an unpublished list of unsolved problems: In which finite groups is the intersection of any two Sylow 2-subgroups normal in at least one of them? For brevity, groups satisfying this condition will be called NI*-groups.In the sequel we consider only finite groups.A group in which the intersection of any two Sylow 2-subgroups is normal in each is called a NI-group.THEOREM. Every NI*-group is a NI-group. If P is a Sylow 2-subgroup of a NI-group and O~(G)= / , then ~r~ I (P) is contained in Z(P) 9The notation is standard.In particular, br21 (P) is the subgroup generated by all elements of order 2 (involutions) in P ; Z(P) is the center of P. 02(G) is the largest normal 2-subgroup of ~ .