### ON CAPABLE GROUPS OF ORDER *p*^{4}

**DOI Number**

https://doi.org/10.22190/FUMI1904633S

**First page**

633

**Last page**

640

#### Abstract

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 the

center of $G$. In a recent paper \cite{2}, the authors

considered 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 $H_7^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\

$H_7^2= \langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{p+1}yz, zy = x^pyz \rangle,$ \\

$H_8^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{-3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\

$H_8^2=\langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{dp+1}yz, zy = x^{dp}yz \rangle$.}

$G$ such that $\frac{G}{Z(G)}~\cong~H$, where $Z(G)$ denotes the

center of $G$. In a recent paper \cite{2}, the authors

considered 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 $H_7^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\

$H_7^2= \langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{p+1}yz, zy = x^pyz \rangle,$ \\

$H_8^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{-3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\

$H_8^2=\langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{dp+1}yz, zy = x^{dp}yz \rangle$.}

#### Keywords

Capable group; p−group; non-abelian p−groups; center

#### Full Text:

PDF#### References

R. Baer: Groups with preassigned central and central quotient groups, Trans.

Amer. Math. Soc. 44 (1938) 387–412.

W. Burnside: Theory of Groups of Finite Order, Cambridge University Press, Cambridge, 1897.

M. Hall and J. K. Senior: The Groups of Order 2 n (n ≤ 6), Macmillan, New York, 1964.

P. Hall: The classification of prime-power groups, J. reine angew. Math. 182 (1940) 130–141.

R. Zainal, N. M. Mohd Ali, N. H. Sarmin and S. Rashid: On the capability of nonabelian groups of order p 4 , Proceedings of the 21st National Symposium on

Mathematical Sciences (SKSM21), AIP Conf. Proc. 1605 (2014) 575–579.

DOI: https://doi.org/10.22190/FUMI1904633S

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