# Find possible subgroups given some elements

• gummz
In summary, the problem presents a subgroup H of the group Z under addition, where H contains elements 2^50 and 3^50. The possibilities for H are Z, 2Z, 10Z, 25Z, and 50Z. However, the notation used for exponents is not consistent with the interpretation for a group operation, and should be interpreted as multiplication rather than repeated addition. Therefore, the possibilities for H are actually Z, 4Z, 8Z, 12Z, and 16Z.
gummz

## Homework Statement

Suppose that H is a subgroup of Z under addition and that H contains 2^50 and 3^50. What are the possibilities for H?

## Homework Equations

Relevant concepts are just the definitions for a group and subgroup.

https://en.wikipedia.org/wiki/Group_(mathematics)

https://en.wikipedia.org/wiki/Subgroup

## The Attempt at a Solution

The solution I was given is the following:

But what I'm wondering about, and would appreciate an answer to, is the following:

But the subgroup has operation "addition" so 2^50 = 50*2 = (2*5*5)*2, and 3^50 = 50*3 = (2*5*5)*3, so the possibilities for H are:

H=Z, H=2Z, H=10Z, H=25Z, H=50Z.

Or am I missing something?

Firstly, after ##"H## contains ##\mathbb{Z}"## you can stop, because ##\mathbb{Z} \subseteq H \subseteq \mathbb{Z}## (the latter for being a subgroup) already implies ##H = \mathbb{Z}##.

What you are missing is, that ##2^{50} \neq 50^2## or ##100##.

Thanks for reply. But if we write out the notation, then, for group with operation "addition":

250 = 50*2, and 502=2*50, and Z is commutative? That is, in general, an=n*a

No, you can't confuse these notations. ##2^{50}## is simply a number and the subgroup is ##(H,+)=2^{50}\cdot (\mathbb{Z},+) + 3^{50}\cdot (\mathbb{Z},+)##. Otherwise one would have defined ##H=100\mathbb{Z}+150\mathbb{Z}=50\mathbb{Z}##.

gummz said:
Thanks for reply. But if we write out the notation, then, for group with operation "addition":
250 = 50*2

I think the notation used in the problem isn't consistent with the interpretation of ##a^k## for a group element ##a##. You are correct that for a group operation denoted "##*##", the interpretation of the notation ##a^3## is usually ##a*a*a##. So for the operation "##+##", the interpretation would be "##a+a+a##". However, I think the problem is using the notation for exponents in the standard sense that it is used for the field of real numbers. By that notation ## 2^3 = (2)(2)(2)## instead of ##2+2+2##.

## What is the purpose of finding possible subgroups?

The purpose of finding possible subgroups is to identify and categorize elements within a larger group that share common characteristics or properties. This can help to better understand the relationships and patterns between elements and can also aid in making predictions or drawing conclusions about the larger group.

## How do you determine the possible subgroups of a given set of elements?

To determine the possible subgroups of a given set of elements, you must first identify any common characteristics or properties among the elements. Then, you can group the elements based on these shared characteristics and analyze the resulting subgroups to see if there are any patterns or relationships that emerge.

## What factors should be considered when identifying subgroups?

When identifying subgroups, factors such as the number of elements, the variety of characteristics among the elements, and the significance of the shared characteristics should be taken into consideration. It is also important to consider the purpose of the subgroup analysis and whether it aligns with the overall goal of the research.

## Are there any tools or techniques that can assist in finding subgroups?

Yes, there are various tools and techniques that can assist in finding subgroups, such as statistical methods like cluster analysis, factor analysis, and discriminant analysis. Additionally, there are software programs and algorithms specifically designed for subgroup analysis.

## Can subgroups change over time?

Yes, subgroups can change over time as new data or information is gathered. This is especially true in dynamic systems or when studying complex phenomena. It is important to regularly reassess and update subgroup analyses to ensure their accuracy and relevance.

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