Mathematics: Give an example of an infinite non abelian group that has exactly six elements of finite order

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Give an example of an infinite non abelian group that

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10.03.2022

Mathematics

Step-by-step explanation:

An infinite non-abelian group that has exactly six elements of finite order is,

Multiplication by concatenation can be done, for example,

$ab \times bc = (ab)(bc) = a b^2 c = ac$ , since b^2 = 1 .

This group is non-commutative, because is not equal to (as this is not a relation in the given presentation).

Now, this is infinite, because we can have 'words' of a, b, c, d, e, f of any length (the only simplifications we can use is when a^2, b^2, c^2, d^2, e^2, f^2 show up).

Therefore, any word (not containing a^2, ..., f^2 ) containing more than one of a,b,c,d,e,f must have infinite order.

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Give an example of an infinite non abelian group that has exactly six elements of finite order

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