دوره 16، شماره 2 - ( 7-1400 )
جلد 16 شماره 2 صفحات 87-73 |
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Shelash H B, Ashrafi A R. The Number of Subgroups of a Given Type in Certain Finite Groups. IJMSI 2021; 16 (2) :73-87
URL: http://ijmsi.ir/article-1-1751-fa.html
URL: http://ijmsi.ir/article-1-1751-fa.html
The Number of Subgroups of a Given Type in Certain Finite Groups. مجله علوم ریاضی و انفورماتیک. 1400; 16 (2) :73-87
چکیده:
The number of subgroups, normal subgroups and characteristic subgroups of a finite group $G$ are denoted by $Sub(G)$, $NSub(G)$ and $CSub(G)$, respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences $U_{6n}$, $V_{8n}$ and $H(n)$ of groups that can be presented as follows:
begin{eqnarray*}
U_{6n} &=& langle a, b mid a^{2n} = b^{3} = e, bab = arangle,
V_{8n} &=& langle a, b mid a^{2n} = b^{4} = e, aba = b^{-1}, ab^{-1}a = brangle,
H(n)&=&langle a,b,c mid a^{2^{n-2}}=b^{2}=c^{2}=e, [x,y]=[y,z]=e, x^{z}=xy rangle.
end{eqnarray*}
For each group, a matrix model containing all information is given.
begin{eqnarray*}
U_{6n} &=& langle a, b mid a^{2n} = b^{3} = e, bab = arangle,
V_{8n} &=& langle a, b mid a^{2n} = b^{4} = e, aba = b^{-1}, ab^{-1}a = brangle,
H(n)&=&langle a,b,c mid a^{2^{n-2}}=b^{2}=c^{2}=e, [x,y]=[y,z]=e, x^{z}=xy rangle.
end{eqnarray*}
For each group, a matrix model containing all information is given.
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