ON CAPABLE GROUPS OF ORDER p4

A group $H$ is said to be capable, if there exists another group$G$ such that $\frac{G}{Z(G)}~\cong~H$, where $Z(G)$ denotes thecenter of $G$. In a recent paper \cite{2}, the authorsconsidered the problem of capability of five non-abelian $p-$groups of order $p^4$ into account. In this paper, we continue this paper by considering three other groups of order $p^4$. It is proved that the group $$H_6=\langle x, y, z \mid x^{p^2}=y^p=z^p= 1, yx=x^{p+1}y, zx=xyz, yz=zy\rangle$$ is not capable. Moreover, if $p > 3$ is prime and $d \not\equiv 0, 1 \ (mod \ p)$ then the following groups are not capable:\\{\tiny $H_7^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\$H_7^2= \langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{p+1}yz, zy = x^pyz \rangle,$ \\$H_8^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{-3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\$H_8^2=\langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{dp+1}yz, zy = x^{dp}yz \rangle$.}

KEMAMPUAN MEMBUAT PETA KONSEP ALUR DARI BUKU FIKSI DAN NON FIKSI SISWA VIII SMP

Abstrak: Keterampilan berbahasa yang paling kompleks adalah keterampilan menulis. Dikatakan kompleks, karena keterampilan menulis menuntut penulis untuk dapat menyusun dan mengorganisasikan isi tulisan serta menuangkan gagasan, perasaan, dalam bentuk bahasa tulis untuk tujuan memberi tahu, meyakinkan dan menghibur para pembaca. Penelitian ini bertujuan untuk mendeskripsikan kemampuan membuat peta konsep alur dari buku fiksi dan non fiksi. Jenis penelitian yang digunakan adalah penelitian deskriptif kuantitatif. Subjek dalam penelitian ini adalah siswa kelas VIII 3 SMP yang berjumlah 33 orang.Metode pengumpulan data yang digunakan dalam penelitian ini yaitu, metode observasi, metode tugas, dan metode dokumentasi.Teknik analisis data menggunakan rumus Penilaian Acuan Patokan (PAP) dengan menentukan kemampuan individu dan kelompok. Berdasarkan hasil penelitian ini, dapat disimpulkan kemampuan membuat peta konsep alur dari buku fiksi dan nonfiksi bahwa kemampuan individu peserta didik yakni, 1) Kelompok berkemampuan tinggi 31 siswa (94%), kelompok berkemampuan sedang 2 siswa (6%), dan kelompok berkemampuan rendah tidak ada (0%). 2) Kemampuan kelompok siswa atau Indeks Prestasi Kumulatif (IPK) dengan nilai 80.78 berada pada kategori tinggi dengan rentang skor 70.5-89. Abstract: The most complex language skills are writing skills. Said to be complex, because writing skills require writers to be able to compile and organize the contents of writing and pour ideas, feelings, in the form of written language for the purpose of informing, convincing and entertaining readers. This study aims to describe the ability to make flow map concepts from fiction and non-fiction books. This type of research is a quantitative descriptive study. The subjects in this study were students of class VIII with 33 people. Data collection methods used in this study are, observation methods, task methods, and documentation methods. Data analysis techniques used the Benchmark Reference Assessment (PAP) formula by determining individual and group abilities. Based on the results of this study, it can be concluded the ability to make flow map concepts from fiction and nonfiction books for class VIII students that the individual abilities of students namely, 1) High-ability groups of 31 students (94%), capable groups moderate 2 students (6%), and low-ability groups are absent (0%). 2) The ability of a student group or Grade Point Average (GPA) with a value of 80.78 is in the high category with a score range of 70.5-89.

Mesomorphic Behavior in Silver(I) N-(4-Pyridyl) Benzamide with Aromatic π–π Stacking Counterions

Organic semiconductor materials composed of π–π stacking aromatic compounds have been under intense investigation for their potential uses in flexible electronics and other advanced technologies. Herein we report a new family of seven π–π stacking compounds of silver(I) bis-N-(4-pyridyl) benzamide with varying counterions, namely [Ag(NPBA)2]X, where NPBA is N-(4-pyridyl) benzamine, X = NO3− (1), ClO4− (2), CF3SO3− (3), PF6− (4), BF4− (5), CH3PhSO3− (6), and PhSO3− (7), which form extended π−π stacking networks in one-dimensional (1D), 2D and 3D directions in the crystalline solid-state via the phenyl moiety, with average inter-ring distances of 3.823 Å. Interestingly, the counterions that contain π–π stacking-capable groups, such as in 6 and 7, can induce the formation of mesomorphic phases at 130 °C in dimethylformamide (DMF), and can generate highly branched networks at the mesoscale. Atomic force microscopy studies showed that 2D interconnected fibers form right after nucleation, and they extend from ~30 nm in diameter grow to reach the micron scale, which suggests that it may be possible to stop the process in order to obtain nanofibers. Differential scanning calorimetry studies showed no remarkable thermal behavior in the complexes in the solid state, which suggests that the mesomorphic phases originate from the mechanisms that occur in the DMF solution at high temperatures. An all-electron level simulation of the band gaps using NRLMOL (Naval Research Laboratory Molecular Research Library) on the crystals gave 3.25 eV for (1), 3.68 eV for (2), 1.48 eV for (3), 5.08 eV for (4), 1.53 eV for (5), and 3.55 eV for (6). Mesomorphic behavior in materials containing π–π stacking aromatic interactions that also exhibit low-band gap properties may pave the way to a new generation of highly branched organic semiconductors.

AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM

Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.

ON THE 2-NILPOTENT MULTIPLIER OF FINITE p-GROUPS

The purpose of this paper is a further investigation on the 2-nilpotent multiplier, $\mathcal{M}$(2)(G), when G is a non-abelian p-group. Furthermore, taking G in the class of extra-special p-groups, we will get the explicit structure of $\mathcal{M}$(2)(G) and will classify 2-capable groups in that class.