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Bachelier
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If a group G is non abelian and a and b in G have orders |a|= n and |b|= m, is there a correlation we can draw between m and n and |ab|?
Deveno said:in general, no. it may be that |ab| isn't even finite.
for example, if G is the quotient of the free group on two generators defined by:
a2 = b2 = e
then ab is not of finite order, even though both a and b are of order 2.
Bachelier said:Thanks. That's what I thought.
So How do I go about figuring out the order of different elements in a group. For instance, consider Dihedral D9 then I can define as : {a,b| |a|=9 and |b|=2}
Now if I'm trying to figure out the different Sylow 2-subgroups, I need to get the elements with order 2. How do I do it?
The order of a non-Abelian group G is the number of elements in the group. It can be found by counting the total number of elements in the group, including the identity element.
The order of an element |a| is the smallest positive integer n such that a^n = e, where e is the identity element. The order of the element |a| must be a factor of the order of the group G.
In a non-Abelian group, the order of the product of two elements |ab| is not necessarily equal to the product of their individual orders |a||b|. However, the order of the product |ab| must divide the least common multiple of the orders |a| and |b|.
No, the orders of two elements |a| and |b| cannot determine the order of their product |ab| in a non-Abelian group. The order of the product |ab| can only be determined by finding the least common multiple of the orders |a| and |b|.
In a non-Abelian group, the order of an element |a| is equal to the order of its inverse a^-1. This is because a and a^-1 have the same order, which is the smallest positive integer n such that a^n = e.