The symmetric genus of alternating and symmetric groups

The symmetric genus of a finite group G has been defined by Thomas W. Tucker as the smallest genus of all surfaces on which G acts faithfully as a group of automorphisms (some of which may reverse the orientation of the surface). This note announces the symmetric genus of all finite alternating and symmetric groups.     

Invariants of the symmetric and alternating groups

A theorem is proved on the growth of the codimension and the defect of the algebras of the invariants of symmetric and alternating groups with the growth of the dimension of the representations without trivial components, containing irreducible, non-one-dimensional, nonstandard subrepresentations. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 201–210, 1987.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

QUASI-PERMUTATION REPRESENTATIONS OF ALTERNATING AND SYMMETRIC GROUPS

The quantities c(G), q(G) and p(G) for finite groups were defined by H. Behravesh. In this article, these quantities for the alternating group An and the symmetric group Sn are calculated. It is shown that c(G) = q(G)=p(G) = n, when G = An or Sn.     

On the probability of generating prosoluble groups

LetF be the free prosoluble group of rankd. We determine the minimum integerk such that the probability of generatingF withk elements is positive.     

Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups Hi, i=1,…,k of the symmetric group Sn (or the alternating group An) such that each element in the group Sn (respectively An) lies in some conjugate of one of the Hi. We prove that this number lies between a?(n) and bn for certain constants a,b, where ?(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+?(n)/2, and we determine the exact value for Sn when n is odd and for An when n is even.     

Normal subgroups of nonstandard symmetric and alternating groups

Let be a nonstandard model of Peano Arithmetic with domain M and let be nonstandard. We study the symmetric and alternating groups S _n and A _n of permutations of the set internal to , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A _n and S _n are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an -valued metric on and (where B _S , B _A are the maximal normal subgroups of S _n and A _n identified earlier) making these groups into topological groups, and by showing that if is -saturated then and are complete with respect to this metric.      

A new characterization of alternating and symmetric groups

In this paper we prove that the alternating groupsA _n , forn=p, p+1, p+2 and symmetric groupsS _n , forn=p, p+1, wherep>=3 is a prime number, can be uniquely determined by their order components. As one of the important consequence of this characterization we show that the simple groupsA _n , wheren=p, p+1, p+2 andp>=3 is prime, satisfy in Thompson's conjecture and Shi's conjecture.     

On Noether's problem for central extensions of symmetric and alternating groups

For a field k and a finite group G acting regularly on a set of indeterminates $\underset{?}{X}={\{X_g\}}_{g\in G}$, let $k(G)$ denote the invariant field $k{(\underset{?}{X})}^G$. We first prove for the alternating group $A_n$ that, if n is odd, then $\mathbb{Q}(A_n)$ is rational over $\mathbb{Q}(A_{n?1})$. We then obtain an analogous result where $A_n$ is replaced by an arbitrary finite central extension of either $A_n$ or $S_n$, valid over $\mathbb{Q}({\zeta }_N)$ for suitable N. Concrete applications of our results yield: (1) a new proof of Maeda's result on the rationality of $\mathbb{Q}{(X_1,\dots ,X_5)}^{A_5}/\mathbb{Q}$; (2) an affirmative answer to Noether's problem over $\mathbb{Q}$ for both $\stackrel{?}{A_5}$ and $\stackrel{?}{S_5}$; (3) an affirmative answer to Noether's problem over $\mathbb{C}$ for every finite central extension group of either $A_n$ or $S_n$ with $n?5$.     

Linear bounds for the normal covering number of the symmetric and alternating groups

Monatshefte für Mathematik - The normal covering number $$gamma (G)$$ of a finite, non-cyclic group G is the minimum number of proper subgroups such that each element of G lies in some...     

Congruences between word length statistics for the finitary alternating and symmetric groups

Bacher and de la Harpe (arxiv:1603.07943, 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and \(p(n)_\mathbf{e }\), a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe (arxiv:1603.07943, 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function \(p_{k}(n)\), we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes \(\ell \ge 5\).     

Explicit determination of generalized symmetric and alternating Galois groups

This paper shows a probabilistic algorithm to decide whether the Galois group of a given irreducible polynomial with rational coefficients is the generalized symmetric group C_p?S_m or the generalized alternating group C_p?A_m. In the affirmative case, we give generators of the group with their action on the set of roots of the polynomial.