On Two Properties of Shunkov Group

One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group $G$ is called Shunkov group if for any finite subgroup $H$ of $G$ in the quotient group $N_G(H)/H$, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group $G$, a situation often arises when it is necessary to move to the quotient group of the group $G$ by some of its normal subgroup $N$. In which cases is the resulting quotient group $G/N$ again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup $N$ is locally finite and the orders of elements of the subgroup $N$ are mutually simple with the orders of elements of the quotient group $G/N$. Let $ \mathfrak{X}$ be a set of groups. A group $G$ is saturated with groups from the set $ \mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $ G$ that is isomorphic to some group of $\mathfrak{X}$ . If all elements of finite orders from the group $G$ are contained in a periodic subgroup of the group $G$, then it is called the periodic part of the group $G$ and is denoted by $T(G)$. It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field. \end{abstracte}

On the nilpotency of the solvable radical of a finite group isospectral to a simple group

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that, except for one specific case, the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

Recognition of symplectic and orthogonal groups of small dimensions by spectrum

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In this paper, we determine all finite groups isospectral to the simple groups [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, we prove that with just four exceptions, every such finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

Quality Of Arguments Used In The First-Round Presidential Debate: Critical Pragmatics And Stephen Toulmin‘s Perspective

It is important for politicians to have a good argumentative skill. For state leaders, the ability to think logically, to use rhetoric, and to argue systematically, scientifically, sharply, and eloquently is very crucial. The year 2019 is the political year for Indonesia. Political campaigns leading to the presidential and the legislative election, both in national and regional levels, will happen in 2019. The focus of this research is to investigate the depth and breadth of the arguments stated by the presidential candidates and how those arguments are presented. The research substantial data source is in the form of speech transcript of the recorded video of the firstround presidential debate by two pairs of president and vicepresident candidates. The research data were the debate arguments found in the video transcript and the contexts surrounding them. The data gathering method used observation by employing recording and note-taking techniques. After the data were gathered, they were selected and classified based on their types for further analysis. The analysis method was distributional method and content analysis. Both data analysis methods were applied to yield significant results of the study. The results showed that there are simple patterns of argument containing claim, subclaim, data, and warrant. The orders of elements of arguments might be varied. The research results also showed that there were various pragmatic meanings found in the arguments used by the president and vice-president candidates. The results of the study which was analyzed using the critical pragmatic perspective reflect how far the candidates were concerned with the marginalized, the underprivileged, and the subjugated people.

CRITICAL PRAGMATIC ANALYSIS OF ARGUMENT OF THE FIRST-ROUND PRESIDENTIAL DEBATE

The results showed that there are simple patterns of argument containing claim, subclaim, data, and warrant (Kneuper, 1978). The orders of elements of arguments might be varied. The research results also showed that there were various pragmatic meanings found in the arguments used by the president and vice-president candidates (Palacio & Gustilo, 2016). The results of the study which was analyzed using the critical pragmatic perspective reflect how far the candidates were concerned with the marginalized, the underprivileged, and the subjugated people.

Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order

In a finite group G, let {\pi_{e}(G)} be the set of orders of elements of G, let {s_{k}} denote the number of elements of order k in G, for each {k\in\pi_{e}(G)} , and then let {\operatorname{nse}(G)} be the unordered set {\{s_{k}:k\in\pi_{e}(G)\}} . In this paper, it is shown that if {\lvert G\rvert=\lvert L_{2}(q)\rvert} and {\operatorname{nse}(G)=\operatorname{nse}(L_{2}(q))} for some prime-power q, then G is isomorphic to {L_{2}(q)} .

On orders of elements of finite almost simple groups with linear or unitary socle

We say that a finite almost simple

On the order of a modulo n, on average

Let [Formula: see text] be an integer. Denote by [Formula: see text] the multiplicative order of [Formula: see text] modulo integer [Formula: see text]. We prove that there is a positive constant [Formula: see text] such that if [Formula: see text], then [Formula: see text] where [Formula: see text] It was known for [Formula: see text] in [P. Kurlberg and C. Pomerance, On a problem of Arnold: The average multiplicative order of a given integer, Algebra Number Theory 7 (2013) 981–999] in which they refer to [F. Luca and I. E. Shparlinski, Average multiplicative orders of elements modulo [Formula: see text], Acta Arith. 109(4) (2003) 387–411.