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Ricardo Andrade
Ricardo Andrade
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Index of $Z(G)$ in the centralizer of an element of a finite 2-group

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that    $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denotedenotes the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

p.s.
I know that ifif $|G|\leq 2^8$ then the equality holds.

Index of $Z(G)$ in the centralizer of an element a finite 2-group

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that  $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

p.s.
I know that if $|G|\leq 2^8$ then the equality holds.

Index of $Z(G)$ in the centralizer of an element of a finite 2-group

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that  $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denotes the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

p.s.
I know that if $|G|\leq 2^8$ then the equality holds.

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Nashenas
Nashenas
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Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

p.s.
I know that if $|G|\leq 2^8$ then the equality holds.

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

p.s.
I know that if $|G|\leq 2^8$ then the equality holds.

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Nashenas
Nashenas
  • 39
  • 3

Index of $Z(G)$ in the centralizer of an element a finite 2-group

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?