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This is a continuation of this questionquestion, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit structure of its centralizer in $\mathrm{GL}_{n}(\mathbb{F}_2)$ (more precisely, of its factor $Z(n)$ over the subgroup generated by $C$), and we computed it. It's an abelian group, and so we were able to compute its decomposition into the sum of cyclic subgroups, as follows, where $n=2^km$, with $m$ odd. $$ Z(n)=\bigoplus_{i=2}^k (\mathbb{Z}/\mathbb{Z}_{2^{k+1-i}})^{2^{i-2}m}\oplus \mathbb{Z}/\mathbb{Z}_{2^{k}}^{m-1}\oplus Z(m). $$ The decomposition of $Z(m)$ is also explicit, and can be described in terms of factors of the $\mathbb{F}_2$-irreducible factors of the polynomial $x^m+1\in\mathbb{F}_2[x]$.

We wonder if this was already done. A comment to loc.cit.loc.cit. says that the case $n=2^k$ was done in in Prop. XI(5.7) of Bass's "Algebraic K-Theory". It is useful to note that the question is equivalent to describing the group of units in the $\mathbb{F}_2$-group algebra of $\mathbb{Z}/n\mathbb{Z}$. It appears that in general the structure of the group of units in the group algebra of $\mathbb{Z}/n\mathbb{Z}$ over a finite field is not known, as one can find fresh publications settling particular cases, cf. e.g. Int.J. Group Theory v.2, No. 4 (2013), 1-6.

This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit structure of its centralizer in $\mathrm{GL}_{n}(\mathbb{F}_2)$ (more precisely, of its factor $Z(n)$ over the subgroup generated by $C$), and we computed it. It's an abelian group, and so we were able to compute its decomposition into the sum of cyclic subgroups, as follows, where $n=2^km$, with $m$ odd. $$ Z(n)=\bigoplus_{i=2}^k (\mathbb{Z}/\mathbb{Z}_{2^{k+1-i}})^{2^{i-2}m}\oplus \mathbb{Z}/\mathbb{Z}_{2^{k}}^{m-1}\oplus Z(m). $$ The decomposition of $Z(m)$ is also explicit, and can be described in terms of factors of the $\mathbb{F}_2$-irreducible factors of the polynomial $x^m+1\in\mathbb{F}_2[x]$.

We wonder if this was already done. A comment to loc.cit. says that the case $n=2^k$ was done in in Prop. XI(5.7) of Bass's "Algebraic K-Theory". It is useful to note that the question is equivalent to describing the group of units in the $\mathbb{F}_2$-group algebra of $\mathbb{Z}/n\mathbb{Z}$. It appears that in general the structure of the group of units in the group algebra of $\mathbb{Z}/n\mathbb{Z}$ over a finite field is not known, as one can find fresh publications settling particular cases, cf. e.g. Int.J. Group Theory v.2, No. 4 (2013), 1-6.

This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit structure of its centralizer in $\mathrm{GL}_{n}(\mathbb{F}_2)$ (more precisely, of its factor $Z(n)$ over the subgroup generated by $C$), and we computed it. It's an abelian group, and so we were able to compute its decomposition into the sum of cyclic subgroups, as follows, where $n=2^km$, with $m$ odd. $$ Z(n)=\bigoplus_{i=2}^k (\mathbb{Z}/\mathbb{Z}_{2^{k+1-i}})^{2^{i-2}m}\oplus \mathbb{Z}/\mathbb{Z}_{2^{k}}^{m-1}\oplus Z(m). $$ The decomposition of $Z(m)$ is also explicit, and can be described in terms of factors of the $\mathbb{F}_2$-irreducible factors of the polynomial $x^m+1\in\mathbb{F}_2[x]$.

We wonder if this was already done. A comment to loc.cit. says that the case $n=2^k$ was done in in Prop. XI(5.7) of Bass's "Algebraic K-Theory". It is useful to note that the question is equivalent to describing the group of units in the $\mathbb{F}_2$-group algebra of $\mathbb{Z}/n\mathbb{Z}$. It appears that in general the structure of the group of units in the group algebra of $\mathbb{Z}/n\mathbb{Z}$ over a finite field is not known, as one can find fresh publications settling particular cases, cf. e.g. Int.J. Group Theory v.2, No. 4 (2013), 1-6.

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Dima Pasechnik
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centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2)

This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit structure of its centralizer in $\mathrm{GL}_{n}(\mathbb{F}_2)$ (more precisely, of its factor $Z(n)$ over the subgroup generated by $C$), and we computed it. It's an abelian group, and so we were able to compute its decomposition into the sum of cyclic subgroups, as follows, where $n=2^km$, with $m$ odd. $$ Z(n)=\bigoplus_{i=2}^k (\mathbb{Z}/\mathbb{Z}_{2^{k+1-i}})^{2^{i-2}m}\oplus \mathbb{Z}/\mathbb{Z}_{2^{k}}^{m-1}\oplus Z(m). $$ The decomposition of $Z(m)$ is also explicit, and can be described in terms of factors of the $\mathbb{F}_2$-irreducible factors of the polynomial $x^m+1\in\mathbb{F}_2[x]$.

We wonder if this was already done. A comment to loc.cit. says that the case $n=2^k$ was done in in Prop. XI(5.7) of Bass's "Algebraic K-Theory". It is useful to note that the question is equivalent to describing the group of units in the $\mathbb{F}_2$-group algebra of $\mathbb{Z}/n\mathbb{Z}$. It appears that in general the structure of the group of units in the group algebra of $\mathbb{Z}/n\mathbb{Z}$ over a finite field is not known, as one can find fresh publications settling particular cases, cf. e.g. Int.J. Group Theory v.2, No. 4 (2013), 1-6.