جستجوی مقالات مرتبط با کلیدواژه « finite group » در نشریات گروه « ریاضی »

Let $G$ be a finite non-trivial group. The intersection graph $\Gamma(G)$, is a graph whose vertices are all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H $ and $K$ if and only if $H\cap K\neq 1$. In this paper, we determine the clique number of the intersection graph of the cyclic groups of orders having at most three primes in their decomposition.

Let $G$ be a finite group. There are several ways to associate a graph to $G$ (see [7] and the references therein). The one we will consider in this paper, is denoted by $\Gamma(G)$ and is called the intersection graph of $G$. The intersection graph $\Gamma(G)$ of a nontrivial group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H $ and $K$ if and only if $H\cap K\neq 1$ where 1 denotes the trivial subgroup of $G$. The graph $\Gamma(G)$ has been extensively studied ( see, for example, [1, 8, 11, 12]). Currently, the present authors in [4], have determined all groups $G$ such that the clique number of $\Gamma(G)$ is less than 5, and also they have given a criterion for solvability of finite groups $G$, by the clique number of $\Gamma(G)$. More precisely, it has been proved that if $G$ is a finite group such that the clique number of $\Gamma(G)$ is less than 13, then $G$ is solvable. Note that 13 is the clique number of $\Gamma(A_5)$, where $A_5$ is the alternating group on 5 letters. Other researches in this topic are intersection graphs of a semigroup, a module, and ideals of a ring, were investigated in [5], [2] and [6, 10], respectively.

We start this section with the following definition:Definition 2.1. The subset $X$ of vertices of a finite graph $\Gamma$ is called a {\it clique}, if the induced subgraph by $X$ is a complete graph. The maximum size of a clique, among all cliques of $\Gamma$, is called the clique number of $\Gamma$ and we denote it by $\omega(\Gamma)$. If $\Gamma$ is empty (without vertex), then we define $\omega(\Gamma)=0$ and $\omega(\Gamma)=1$ if $\Gamma$ is null (with a non-empty vertex set with no edges). A clique $X$ in $\Gamma$ is called {\it maximal} if there is no clique $Y$ in $\Gamma$ such that $X\subsetneq Y$. Note that the maximum size, among all maximal cliques in $\Gamma(G)$, is $\omega(\Gamma(G))$. To prove the main theorems, we need the following result that its proof can be found in the most valid book of group theory. Proposition 2.2. If $G=\langle g\rangle$ is a cyclic group of ordsr $n$, then for any divisor $s$ of $n$, there is a unique subgroup $H=\langle g^{\frac{n}{s}}\rangle$ of $G$ of order $s$. The following result is a consequence of the above proposition. Corollary 2.3. Let $G$ be a finite cyclic group. Then, the intersection of two proper subgroups of $G$ is non-trivial if and only if their orders are not relatively prime. For a natural number $n$, we denote by $C_n$ the cyclic group of order $n$, $\pi(n)$ the set of prime divisors of $n$ and $d(n)$ the number of all divisors of $n$. Note that if $p$ is a prime number and $n$ is a multiple of $p$, then the number of divisors of $n$ with multiple $p$ is $d(\frac{n}{p})$. If $V(\Gamma(G))$ is the set of vertices of $\Gamma(G)$, then by Proposition 2.2, we have $|V(\Gamma(C_n))|=d(n)-2$. In this paper, we obtain $\omega(\Gamma(C_n))$, where $|\pi(n)|=3$.

Now, we state and prove our main results. First, we find the clique number of a cyclic group of a prime power order. Proposition 3.1. If $p$ is a prime and $m$ is a positive integer, then $\omega(\Gamma(C_{p^m}))=m-1$.Proof. Since $|V(\Gamma(C_n))|=d(n)-2$ and $d(p^m)=m+1$, we get the conclusion. In the sequel, assume that $p_1, p_2$ and $p_3$ are distinct primes. Also assume that $n_1, n_2$ and $n_3$ are positive integers such that $n_1\geq n_2\geq n_3$. In the following results, we obtain the clique number of the intersection graph of group $C_{p_1^{n_1}p_2^{n_2}}$. We recall that $d(p_1^{n_1}p_2^{n_2})=(n_1+1)(n_2+1)$. Proposition 3.2. We have$\omega(\Gamma(C_{p_1^{n_1}p_2^{n_2}}))=d(p_1^{n_1}p_2^{n_2})-2-d(p_2^{n_2})=n_1n_2+n_1-1$. Proof. Suppose that $G=C_{p_1^{n_1}p_2^{n_2}}$. Then, we define the subsets of $V(\Gamma(G))$ as follows:♦ For $1\leq i\leq 2$, $V_{p_i}(\Gamma(G))$ is the set of all   proper subgroups $H$ of $G$ such that $\pi(|H|)=\{p_i\}$.
♦ $V_{p_1p_2}(\Gamma(G))$ is the set of all proper subgroups $H$ of $G$ such that $\pi(|H|)=\{p_1, p_2\}$. It is clear that $\{V_{p_1}(\Gamma(G)), V_{p_2}(\Gamma(G)), V_{p_1p_2}(\Gamma(G)) \}$ forms a partition for $V(\Gamma(G))$. By Proposition 2.2, we have $|V_{p_i} (\Gamma(G))|=d(p_i^{n_i})-1=n_i$ and $|V_{p_1p_2}(\Gamma(G))|=d(\frac{n}{p_1p_2})-1=n_1n_2-1$. By Corollary 2.3, in $\Gamma(G)$, each element of the clique $V_{p_1}(\Gamma(G))$ does not join to any element of the clique $V_{p_2}(G)$ and moreover all elements of $V_{p_i}(\Gamma(G))$ for $i=1, 2$ join to all elements of the clique $V_{p_1p_2}(G)$. Therefore $V_{p_1}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))$ and $V_{p_2}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))$ are the only maximal cliques of $\Gamma(G)$ and since $n_1\geq n_2$, we have the result.Now we state the last main result. Theorem 3.3. Let $G= C_n$ where $n=p_1^{n_1}p_2^{n_2}p_3^{n_3}$. Then\\(1) If $n_1\geq n_2n_3$, then $\omega(\Gamma(G))=d(\frac{n}{p_1})-1=n_1+n_1n_2+n_1n_3+n_1n_2n_3-1$.\\(2) If $n_1\leq n_2n_3$, then $\omega(\Gamma(G))=d(\frac{p_1^{n_1}p_2^{n_2}}{p_1p_2})+d(\frac{p_1^{n_1}p_3^{n_3}}{p_1p_3})+d(\frac{p_2^{n_2}p_3^{n_3}}{p_2p_3})+d(\frac{n}{p_1p_2p_3})-1$\\$$\hspace{5cm}=n_1n_2+n_1n_3+n_2n_3+n_1n_2n_3-1.~~~~~~~~~~~~~~~~~~~~~~~\hspace{4cm}$$Proof. Similar to the proof of Proposition 3.2, we define the subsets of $V(\Gamma(G))$ as follows:\\♦ For $1\leq i\leq 3$, $V_{p_i}(\Gamma(G))$ is the set of all subgroups $H$ of $G$ such that $\pi(|H|)=\{p_i\}$. Therefore, $|V_{p_i}(\Gamma(G))|=d(p_i^{n_i})-1=n_i$.♦ $V_{p_ip_j}(\Gamma(G))$ is the set of all subgroups $H$ of $G$ such that $\pi(|H|)=\{p_i, p_j\}$ for $1\leq i<j\leq 3$. Therefore, $|V_{p_ip_j}(\Gamma(G))|=d(\frac{p_i^{n_i}p_j^{n_j}}{p_ip_j})=n_in_j$where $i\neq j$.♦ $V_{p_1p_2p_3}(\Gamma(G))$ is the set of all proper subgroups $H$ of $G$ such that $\pi(|H|)=\{p_1, p_2, p_3\}$. So, we have$|V_{p_1p_2p_3}(\Gamma(G))|=d(\frac{n}{p_1p_2p_3})-1=n_1n_2n_3-1$.By Proposition 2.2, the above sets forms a partition for $V(\Gamma(G))$. By Corollary 2.3, in $\Gamma(G)$, each element of the clique $V_{p_i}(\Gamma(G))$ does not join to any element of the clique $V_{p_j}(G)\cup V_{p_jp_k}(G)$ for all distinct $i, j, k$ and moreover all elements of $V_{p_i}(\Gamma(G))$ for $i=1, 2, 3$, join to all elements of the clique $V_{p_1p_2p_3}(G)\cup V_{p_ip_j}(G)$, where $1\leq i\neq j\leq 3$. Since $|G|$ has three prime divisors, the intersection of every two subgroups of $G$ of orders with two distinct prime divisors, is non-trivial (by Corollary 2.3). It follows that $V_{p_1p_2}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))$ is a cloque in $\Gamma(G)$. Now, we define $W_i$ as follows for $1\leq i\leq 4$: $$W_1=V_{p_1}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_1|=n_1+n_1n_2+n_1n_3+n_1n_2n_3-1$$ $$W_2=V_{p_2}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_2|=n_2+n_1n_2+n_2n_3+n_1n_2n_3-1$$ $$W_3=V_{p_3}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_3|=n_3+n_1n_3+n_2n_3+n_1n_2n_3-1$$ $$W_4= V_{p_1p_2}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_4|=n_1n_2+n_1n_3+n_2n_3+n_1n_2n_3-1.$$ It is easy to see that $W_1, W_2, W_3$ and $W_4$ are the only maximal cliques in $\Gamma(G)$. Therefore, $$\omega(\Gamma(G))=max\{|W_1|, |W_2|, |W_3|, |W_4|\}.$$Since $n_1\geq n_2\geq n_3$, we have $|W_1|\geq |W_2|\geq |W_3|$. Thus$$\omega(\Gamma(G))=max\{|W_1|, |W_4|\}=max\{n_1+n_1n_2+n_1n_3+n_1n_2n_3-1, n_1n_2+n_1n_3+n_2n_3+n_1n_2n_3-1\},$$this completes the proof.

One of the most important topics in the theory of fuzzy groups is the classification of fuzzy subgroups of a finite group. Enumeration of fuzzy subgroups has been done in the theory of fuzzy groups using the equivalence relation defined by Murali and Makamba. Using this idea, Neeraj and Sharma enumerated Intuitionistic fuzzy subgroups of a finite Abelian group by concept of double pinned flags.We are in thisWe do this in another way for some finite Abelian and non-Abelian groups. In fact, with the help of a suitable equivalence relation on Intuitionistic fuzzy subgroups and the lattice of subgroups, we obtain the enumeration of Intuitionistic fuzzy subgroups of finite groups of order 12. Also, we obtain the enumeration of Intuitionistic fuzzy subgroups of finite groups $Z_{p^k}\times Z_q$ and $Z_{p^k}$ with a different method from the article [9]. Finally, with the help of the results obtained in this paper, we present the number of Intuitionistic fuzzy subgroups on a group with order less than 16 in a table.

The Camina triple condition is a generalization of the Camina condition in the theory of finite groups. The irreducible characters of Camina triples have been verified in the some special cases. In this paper, we consider a Camina triple (G,M,N)  and determine the irreducible characters of G in terms of the irreducible characters of M and G/N../files/site1/files/61/0Abstract(1).pdf