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Register a new userSome examples of invariably generated groups
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Ashot Minasyan
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A group $G$ is invariably generated (IG) if there is a subset $S subseteq G$ such that for every subset $S subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S$ generates $G$. $G$ is finitely invariably generated (FIG) if, in addition, one can choose such a subset $S$ to be finite. In this note we construct a FIG group $G$ with an index $2$ subgroup $N lhd G$ such that $N$ is not IG. This shows that neither property IG nor FIG is stable under passing to subgroups of finite index, answering questions of Wiegold and Kantor, Lubotzky, Shalev. We also produce the first examples of finitely generated IG groups that are not FIG, answering a question of Cox.
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It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural generalized Ramanujan congruences, as we call them, for which we give numerical evidence in Appendix C.
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The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_infty$-property and are not co-Hopfian for $n ge 3$.
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 different 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 $U_5(2)$.
The degree pattern of a finite group is the degree sequence of its prime graph in ascending order of vertices. We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise nonisomorphic finite groups with the same order and degree pattern as the group under consideration. In this article the problem of OD-characterization is solved for some simple unitary groups. It was shown, in particular, that the simple unitary groups $U_3(q)$ and $U_4(q)$ are OD-characterizable, where $q$ is a prime power $<10^2$.
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