# Thread: find an integer k such that the subgroup is <k>

1. ## find an integer k such that the subgroup is <k>

The smallest subgroup containing a collection of elements S is the
subgroup H with the property that if K is any subgroup containing
S then K also contains H. (So, the smallest subgroup containing S is
contained in every subgroup that contains S.) The notation for this
subgroup is <S>. In the group Z, find
a. <8, 14>
b. <8, 13>
c. <6, 15>
d. <m, n>
e. <12, 18, 45>.
In each part, find an integer k such that the subgroup is k.

2. ## Re: find an integer k such that the subgroup is <k>

Originally Posted by MathGeni04
The smallest subgroup containing a collection of elements S is the
subgroup H with the property that if K is any subgroup containing
S then K also contains H. (So, the smallest subgroup containing S is
contained in every subgroup that contains S.) The notation for this
subgroup is <S>. In the group Z, find
a. <8, 14>
So the smallest set of integers containing 8 and 14? That would, of course, include 0 as well as 8+ 8= 16 and, in fact, any multiple of 8 or 14. It would include 8+ 14= 22, 14- 8= 6, etc. In fact it would include any x= 8n+ 14m= 2(4n+ 7m) for any integers n and m. (So, first, it must include only even integers.) What can you say about numbers of the form 4n+ 7m? One of the results of the "Euclidean Algorithm" is that since 4 and 7 are "relatively prime", there exist m and n such that 4n+ 7m= x for all integers x.

b. <8, 13>
8 and 13 are relatively prime.

c. <6, 15>
6= 3(2), 15= 3(5) so 6m+ 15n= 3(2m+ 5n).

d. <m, n>
Is no information at all given about m and n?

[quote]e. <12, 18, 45>.[quote]
12= 2(2)(3), 18= 2(3)(3), 45= 3(3)(5). The greatest common divisor is 3.

In each part, find an integer k such that the subgroup is k.
You mean "the subgroup is <k>."

3. ## Re: find an integer k such that the subgroup is <k>

Hi,
The attachment expands a little on the previous response. I hope if you read and understand this, you can answer all your questions easily. For the last question e, you just have to note that <a,b,c>=<<a,b>,c>.

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# In the group z, find <8,14>. In each part, find an integer k such that the subgroup is <k>

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