2016年2月24日 · In my opinion, precise definition of a free group should be based on the concepts word, sub-word, empty word, inverse word, cancelation, reduced word, and …

This starts with the "standard" construction using finite strings over X = X ∪ X ′ then asks for an opinion on quotienting by some equivalence relation. Can a free group over a set be …

2017年7月25日 · Now, the free group on two generators describes exactly the difficulty for a small insect to walk on the contour of the petals (no flying's allowed!). This "difficulty" is formally …

So, free groups, in this sense, are so general, that every group is 'contained' (as a quotient) of them. On the other hand, every group can be embedded to a symmetric group.

2019年12月22日 · The free group is interesting because it is the most general possible group on a given set S S, with no assumption of any relations. It's useful for constructing groups, e.g., via …

2016年4月11日 · A free group is indeed infinite, I don't see any problem here. For the inverse, note that what you said is not the complete definition of the free group. You have to take the …

2023年4月30日 · In Johnson's 'Topics in the Theory of Group Presentations', one can find this theorem after the definition of free groups using the universal property. Theorem. Free groups …

2017年8月6日 · The free group Z ∗Z Z ∗ Z is basically an infinite non abelian gorup with each of its elements having infinite order except identity. I can't think of any other group which can be …

Every finitely generated free group is a subgroup of F2 F 2, the free group on two generators. This is an elementary fact, as is the fact that G G, finitely presented, is the quotient of F(|S|) F (| S |) …

The free product Z ∗ Z, also called the free group on two generators, is the group consisting of all words like a4b − 2a15b3⋯, where you multiply by concatenating words followed by combining …