• Journal of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.227-235
• /
• 1997

In recent years there has been much interest in the study of groups satisfying various permutability conditions (see, for instance, [1], [2] and [3]). More recently, the following condition has been studied: for some , if S is any subset of m elements of a group G, then $$\mid$S^2$\mid$ < m^2$ (where, for subsets A, B of G, AB stands for ${ab; a \in A, b \in B}$).

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.1
• /
• pp.225-234
• /
• 2021

Let D be a division ring and n > 1 be an integer. In this paper, it is shown that if D ≠ ��3, then every subpermutable subgroup of the general skew linear group GLn(D) is normal. By applying this result, we show that every subpermutable subgroup of the unit group (ℝG)∗ of the real group algebras RG of finite groups G is normal in (ℝG)∗.

• Journal of applied mathematics & informatics
• /
• v.20 no.1_2
• /
• pp.75-96
• /
• 2006

Let m, n be positive integers. We denote by R(m,n) (respectively P(m,n)) the class of all groups G such that, for every n subsets $X_1,X_2\ldots,X_n$, of size m of G there exits a non-identity permutation $\sigma$ such that $X_1X_2{\cdots}X_n{\cap}X_{\sigma(1)}X_{/sigma(2)}{\cdots}X_{/sigma(n)}\neq\phi$ (respectively $X_1X_2{\cdots}X_n=X_{/sigma(1)}X_{\sigma(2)}{\cdots}X_{\sigma(n)}$). Let G be a non-abelian group. In this paper we prove that (i) $G{\in}P$(2,3) if and only if G isomorphic to $S_3$, where $S_n$ is the symmetric group on n letters. (ii) $G{\in}R$(2, 2) if and only if ${\mid}G{\mid}\geq8$. (iii) If G is finite, then $G{\in}R$(3, 2) if and only if ${\mid}G{\mid}\geq14$ or G is isomorphic to one of the following: SmallGroup(16, i), $i\in$ {3, 4, 6, 11, 12, 13}, SmallGroup(32, 49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of order m in the GAP [13] library.