On the commutator lengths of certain classes of finitely presented groups
For a finite group G=〈X〉 (X≠G), the least positive integer MLX(G) is called the maximum length of G with respect to the generating set X if every element of G may be represented as a product of at most MLX(G) elements of X. The maximum length of G, denoted by ML(G), is defined to be the minimum of {...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2006-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/74981 |
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| Summary: | For a finite group G=〈X〉 (X≠G), the least
positive integer MLX(G) is called the maximum length of G
with respect to the generating set X if every element of G may
be represented as a product of at most MLX(G) elements of X.
The maximum length of G, denoted by ML(G), is defined to be
the minimum of {MLX(G)|G=〈X〉,X≠G,X≠G−{1G}}. The well-known commutator length of a group
G, denoted by c(G), satisfies the inequality c(G)≤ML(G′), where G′ is the derived subgroup of G. In this paper
we study the properties of ML(G) and by using this inequality
we give upper bounds for the commutator lengths of certain classes
of finite groups. In some cases these upper bounds involve the
interesting sequences of Fibonacci and Lucas numbers. |
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| ISSN: | 0161-1712 1687-0425 |