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A CHARACTERIZATION OF THE UNIT GROUP IN ℤ[T×C<sub>2</sub>]
Bilgin, Tevfik,Kusmus, Omer,Low, Richard M. Korean Mathematical Society 2016 대한수학회보 Vol.53 No.4
Describing the group of units $U({\mathbb{Z}}G)$ of the integral group ring ${\mathbb{Z}}G$, for a finite group G, is a classical and open problem. In this note, we show that $$U_1({\mathbb{Z}}[T{\times}C_2]){\sim_=}[F_{97}{\rtimes}F_5]{\rtimes}[T{\times}C_2]$$, where $T={\langle}a,b:a^6=1,a^3=b^2,ba=a^5b{\rangle}$ and $F_{97}$, $F_5$ are free groups of ranks 97 and 5, respectively.
A characterization of the unit group in $\mathbb{Z}[T \times C_2]$
Tevfik Bilgin,Omer Kusmus,Richard M. Low 대한수학회 2016 대한수학회보 Vol.53 No.4
Describing the group of units $U(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$, for a finite group $G$, is a classical and open problem. In this note, we show that $U_1(\mathbb{Z}[T \times C_2]) \cong [F_{97} \rtimes F_5] \rtimes [T \times C_2]$, where $T = \langle a, b: a^6 = 1, a^3 = b^2, ba = a^5b \rangle$ and $F_{97}$, $F_5$ are free groups of ranks 97 and 5, respectively.
CHARACTERIZATION OF CENTRAL UNITS OF ℤA<sub>n</sub>
Bilgin, Tevfik,Gorentas, Necat,Kelebek, I. Gokhan Korean Mathematical Society 2010 대한수학회지 Vol.47 No.6
The structure of V(Ƶ($ZA_n$)) is known when $n\leq6$. If n = 5 or 6, then a complete set of generators of V (Ƶ($ZA_n$)) has been deter-mined. In this study, it was shown that V (Ƶ($ZA_n$)) is trivial when n = 7, 8 or 9 and it is generated by a single unit u when n = 10 or 11: This unit u is characterized explicitly for n = 10 or 11 by using irreducible characters of $A_n$.
CHARACTERIZATION OF CENTRAL UNITS OF ZAn
Tevfik Bilgin,Necat Gorentas,I. Gokhan Kelebek 대한수학회 2010 대한수학회지 Vol.47 No.6
The structure of V (Z(ZAn)) is known when n ≤ 6. If n = 5or 6, then a complete set of generators of V (Z(ZAn)) has been deter-mined. In this study, it was shown that V (Z(ZAn)) is trivial when n = 7; 8 or 9 and it is generated by a single unit u when n = 10 or 11: This unit u is characterized explicitly for n = 10 or 11 by using irreducible characters of An.