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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Isomorphisms of graph groups


Author: Carl Droms
Journal: Proc. Amer. Math. Soc. 100 (1987), 407-408
MSC: Primary 20F05; Secondary 05C25, 20F12
DOI: https://doi.org/10.1090/S0002-9939-1987-0891135-1
MathSciNet review: 891135
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Abstract: Given a graph $ X$, define the presentation $ PX$ to have generators the vertices of $ X$, and a relation $ xy = yx$ for each pair $ x,y$ of adjacent vertices. Let $ GX$ be the group with presentation $ PX$, and given a field $ K$, let $ KX$ denote the $ K$-algebra with presentation $ PX$. Given graphs $ X$ and $ Y$ and a field $ K$, it is known that the algebras $ KX$ and $ KY$ are isomorphic if and only if the graphs $ X$ and $ Y$ are isomorphic. In this paper, we use this fact to prove that if the groups $ GX$ and $ GY$ are isomorphic, then so are the graphs $ X$ and $ Y$.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0891135-1
Article copyright: © Copyright 1987 American Mathematical Society