By Honsberger R.

Selecting his choices to be comprehensible to the common university freshman, Honsberger (combinatorics and optimization, U. of Waterloo, Canada) provides mathematical difficulties and proofs from Euclidean geometry, combinatorial geometry, algebra, and quantity thought

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Group Theory 5, 383–401, 2002. [9] L´aszl´o Babai, P´eter P. P´ alfy and Jan Saxl. On the number of p-regular elements in ﬁnite simple groups. LMS J. Comput. Math. 12, 82–119, 2009. ´ [10] L´ aszl´o Babai, Robert Beals and Akos Seress. Polynomial-time Theory of Matrix Groups. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, pages 55–64, 2009. O’Brien: Algorithms for matrix groups 319 [11] Henrik B¨ aa¨rnhielm. Algorithmic problems in twisted groups of Lie type.

P. Glasby and Cheryl E. Praeger. Towards an eﬃcient Meat-axe algorithm using f -cyclic matrices: The density of uncyclic matrices in M(n, q). J. Algebra 322, 766– 790, 2009. [50] Daniel Gorenstein, Richard Lyons and Ronald Solomon. The classiﬁcation of the ﬁnite simple groups. Number 3. American Mathematical Society, Providence, RI, 1998. [51] Robert Guralnick, Tim Penttila, Cheryl E. Praeger and Jan Saxl. Linear groups with orders having certain large prime divisors. Proc. London Math. Soc. 78, 167–214, 1999.

Xk , t | xt1 = φ(x1 ), . . , xtk = φ(xk ) for some injective endomorphism φ of Fk . For example, HT = x, y, t | txt−1 = xy, tyt−1 = yx , so the endomorphism φ is given by φ : x → xy, y → yx. Sapir: Residual properties of 1-relator groups 334 It is easy to see that every element in G has the form tk w(x1 , . . , xk )tl . If k +l = 0, then the element survives the homomorphism xi → 0, t → 1, G → Z. Hence we can assume that k + l = 0, and the element is conjugate to w(x1 , . . , xk ). Thus it is enough to consider elements from Fk only.