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- Thread starter Bachelier
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Deveno

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for example, if G is the quotient of the free group on two generators defined by:

a

then ab is not of finite order, even though both a and b are of order 2.

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Thanks. That's what I thought.

for example, if G is the quotient of the free group on two generators defined by:

a^{2}= b^{2}= e

then ab is not of finite order, even though both a and b are of order 2.

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?

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Deveno

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for the dihedral group DThanks. 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?

if n is odd, then the sylow 2-subgroups are all of order 2, because half the elements are of odd order (the rotations). geometrically, it's obvious the reflections are of order 2, but we can prove this algebraically, as well:

given: r

first, a small detour:

lemma: rs = sr

for k = 1, this is given by the defining relations on D

suppose this is true for k = m.

then r

= (sr

so the general result holds by induction on k.

corollary: sr

since r

we have r

ok, back to showing what the square of a reflection is:

(sr

= (r

= r

so every reflection has order 2. now, since 9 is odd, this means we have 9 sylow 2-subgroups, one for each reflection.

(if n is even, we have a slightly more involved situation, because now 4 divides the order of the group).

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