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Primitive Elements in Free Groups

Martin J. Evans
Proceedings of the American Mathematical Society
Vol. 106, No. 2 (Jun., 1989), pp. 313-316
DOI: 10.2307/2048805
Stable URL: http://www.jstor.org/stable/2048805
Page Count: 4
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Primitive Elements in Free Groups
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Abstract

Let Fn denote the free group of rank n generated by x1, x2,..., xn. We say that y ∈ Fn is a primitive element of Fn if it is contained in a set of free generators of Fn. In this note we construct, for each integer n ≥ 4, an (n - 1)-generator group H that has an n-generator, 2-relator presentation $H = \langlex_1,\ldots, x_n\midr_1, r_2\rangle$ such that the normal closure of $\{r_1, r_2 \}$ in Fn does not contain a primitive element of Fn.

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