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Find the Automorphism group of a Brandt semigroup $B(G,2)$ ,where $G $ is a cyclic group of order 4.

Take $G=\{ e, a ,a^2, a^3\}$

$B(G,2) = \{ (i,a^s , j) : 1 \leq i,j \leq 2 \ \ , 0\leq s \leq 3 \} \cup \{0\}$ and the binary operation is defind by $$(i, a^r , j) (k,a^s , l) = \begin{cases} (i,a^ra^s,l) & \text{if } \ \ j=k \\ 0 &\text{if } otherwise \end{cases}$$

user120386
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  • This is not really research level. The endomorphisms of any Rees matrix semigroup are described in any standard text and yours is particularly easy. – Benjamin Steinberg Sep 29 '16 at 12:00
  • Basically i am interested to find the Automorphism group of $B(G,n)$, where $G$ is any group. I know that the $B_n $ is isomorphic to $S_n$. I want to gneralize this result. – user120386 Sep 29 '16 at 13:50
  • The generalization you want can be found in standard semigroup books. You can conjugate by permutation matrices and apply group automorphisms and combine these two operations. Thats it. – Benjamin Steinberg Sep 29 '16 at 14:56

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As Benjamin Steinberg remarked, this special case is not very complicated and does not need my answer. However the automorphism group of $B(S,n)$, where $S$ is a monoid and $n\in\mathbb{N}$, was completely determined in Gutik (see Theorem 2).

khers
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