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March 30th, 2012, 07:01 PM   #1
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Group and subgroup ordering

Hey,

I'm just trying to grasp ordering of groups and subgroups a little better,

I get the basics of finding the order of elements knowing the group but I have a few small questions,

If you have a group of say, order 100, what would the possible orders of an element say g^12 in the group be?

Would they just be the multiples of g^12 that have power less then 100? Like 2,3,4,5,6,7,8 ?

I read up on Lagrange's theorem but from my understanding that is related to the order of subgroups not individual group elements.

Which brings me to my next question,

Say if you have a group with two subgroups A and B, where the order of A is 120 and the order of B is say 105,

The lowest common multiple of these two numbers is 840, so would that be the order of the group? Since both the orders of the subgroups need to be a multiple of the order of the group?

But if you take the intersection between the two subgroups A and B, what would be the possible orders of the intersection?

Would one of them be 120-105 = 15? If all of the elements in B are also in A.

Thanks,
Linda
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March 31st, 2012, 12:39 PM   #2
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Re: Group and subgroup ordering

Quote:
If you have a group of say, order 100, what would the possible orders of an element say g^12 in the group be?
The set defined by is a subgroup, so it can only be a divisor of the order of the group, so of 100.
Quote:
The lowest common multiple of these two numbers is 840, so would that be the order of the group?
At least should be the minimum order possible.
Quote:
But if you take the intersection between the two subgroups A and B, what would be the possible orders of the intersection?
The maximum order has to be the greatest common divisor since the intersection of these two subgroups is a subgroup. Then it can be any divisor of this number, it could very well just be the identity element...
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