Free Groups Are Not Boundedly Generated
Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups.
Since for any n ≥ 2, the free group on 2 generators F2 contains the free group on n generators Fn as a subgroup of finite index (in fact n – 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true for all of them. Similarly, since SL2(Z) contains F2 as a subgroup of index 12, it is enough to consider SL2(Z). In other words, to show that no Fn with n ≥ 2 has bounded generation, it is sufficient to prove this for one of them or even just for SL2(Z) .
Read more about this topic: Boundedly Generated Group
Famous quotes containing the words free, groups and/or generated:
“The poet will prevail to be popular in spite of his faults, and in spite of his beauties too. He will hit the nail on the head, and we shall not know the shape of his hammer. He makes us free of his hearth and heart, which is greater than to offer one the freedom of a city.”
—Henry David Thoreau (1817–1862)
“And seniors grow tomorrow
From the juniors today,
And even swimming groups can fade,
Games mistresses turn grey.”
—Philip Larkin (1922–1986)
“It is precisely the purpose of the public opinion generated by the press to make the public incapable of judging, to insinuate into it the attitude of someone irresponsible, uninformed.”
—Walter Benjamin (1892–1940)