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I will start by apologizing as many will not like this question.

I am reading the paper COHOMOLOGY THEORY OF GROUPS WITH A SINGLE DEFINING RELATION and having focused on typology throughout my studies i finding myself to be in a lost notation wise.

  1. Let $F$ be a free subgroup in $G$ with generators $x_i$ , then elements in $F$ are represented by "words" or a final sequence of $x_i^{\pm1}$. In Page 650 (first one in this article) and 658 they state that every such word $R$ "can be expressed uniquely as a power $R=Q^q$ for $q$ maximal" Nowhere do they explain what is this $Q$, any ideas, links, resources for me to search in?

  2. Let $R$ be a normal subgroup in $G$, what does the notation $(R, R)$ symbolize? they used it in pages 650 and 658 without ever explaining.

Am i reading this paper wrong? Or are these notation so common they are never introduced?

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1. $R=xyz$ can be written as $R=(xyz)^1$, so here $Q=xyz$. $R=xyxy$ can be written as $R=(xy)^2$, so here $Q=xy$. 2. Is probably the commutator. –  user641 Dec 1 '12 at 21:38
Oh, that makes sense, thank you very much, What about (R,R)? –  user44874 Dec 1 '12 at 21:42
The general references for one-relator groups are two books, both called "combinatorial group theory" (the second was named in honour of the first!). The first was by Magnus, Karrass and Solitar, the second by Lyndon (he who wrote the paper you are reading) and Schupp. I suppose they're pretty out of date now, but they are excellent places to start: the tools laid down by Magnus in his 1931 thesis are still used today! –  user1729 Dec 3 '12 at 10:24

1 Answer 1

  1. Maybe you prefer more elaborate would be a formulation like this: "For every word $R$ over the alphabet $A$ there exist $(Q,q)$ such that $Q$ is a word over the alphabet $A$ and $q\in\mathbb N$ and $R=Q^q$ is the $q$-fold concatenation of $Q$ with itself. One example of such a pair for arbitray $R$ is $(R,1)$ as trivially $R=R^1$. If $R$ is not the empty word, then $Q$ in such a pair cannot be the empty word and hence $q$ is bounded from above by the length of the word $R$. Hence there exist such pairs $(Q,q)$ for which $q$ attains te maximal possible value. Since $Q_1\ne Q_2$ implies $Q_1^q\ne Q_2^q$ (if $q\ge 1$), the word $Q$ for which the maximal possible $q$ is attained, is uniquely determined."

  2. Since they refer in passing to $R/(R,R)$ as the abelianized group, $(R,R)$ should denote the commutator. Today, writing $[R,R]$ seems to be the preferred notation.

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