Group Multiplication: HH vs H - Is there a Difference?

In summary, the book claims that HH=H because H is a subgroup, but this is not true because HH is not an ordered pair.
  • #1
pivoxa15
2,255
1
Let G be a group and H a subgroup of G.

The book claims HH=H because H is a subgroup.

Group multiplication is defined as AB={(a,b): a in A, b in B}

So HH should be ordered pairs with each pair containing two identical elements in H. But why is the answer H, which is not an ordered pair?

I think they have used this definition http://en.wikipedia.org/wiki/Product_of_subgroups instead of http://en.wikipedia.org/wiki/Direct_product_(group_theory)

Are the two completely different? The latter they direct product. If one write HH does it not refer to direct product? I always thought not putting a sign such as X means the same thing as putting X. Or is this convention only for elements of a group. So when doing operations on whole groups, putting or not putting a sign has different consequences?
 
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  • #2
[tex]HH = \left\{ h_{1}h_{2}\ |\ h_{1}, h_{2} \in H \right\}[/tex]
[tex]h_{1}h_{2} \in H[/tex] since H is a subgroup and hence closed
therefore, [tex]HH \subseteq H[/tex]

now let [tex]h \in H[/tex]
[tex]h = he \in HH[/tex] where e is the identity element
since [tex] h \in HH, H \subseteq HH[/tex]

we have [tex]HH \subseteq H[/tex] and [tex]H \subseteq HH[/tex]
therefore,
[tex]HH = H[/tex]
 
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  • #3
I understand that. To show equality between two relations, do I always have to prove if and only if?

I take it that direct product is entirely different to product of subgroups? Couldn't you have a direct product between subgroups?
 
  • #4
pivoxa15 said:
Group multiplication is defined as AB={(a,b): a in A, b in B}
that is the definition of cartesian product, isn't it? here HH is not a cartesian product
 
  • #5
pivoxa15 said:
I take it that direct product is entirely different to product of subgroups? Couldn't you have a direct product between subgroups?

They are groups, so of course you can have direct products. However, if H and K are subgroups of G, then in general HxK will not be a subgroup of G.
 
  • #6
So as a general rule, always use X when meaning direct product between groups and no symbol when meaning subgroup multiplication which is not in Cartesian coordinate space. Correct?

Also with the proof when assuming product of subgroups what about

H is a subgroup of G.
[tex]H=\left\{h_{1}, h_{2}, ..., h_{n}\}[/tex]

[tex]HH = \left\{ h_{1}h_{1}, h_{2}h_{2}, ..., h_{n}h_{n}\}[/tex]

use close under multiplication for products of elements in subgroups.

[tex]HH = \left\{ h_{1}, h_{2}, ..., h_{n}\}

= H[/tex]
 
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  • #7
Closure only tells you that HH is a subset of H. From what you have written, you apparently are saying that the elements of HH are of the for hh for h in H. You're set sould be, if you want to needlessly introduce subscripts:

[tex] \{ h_i h_j : 1 \le i,j \le n\}[/tex]

not, as you have written

[tex] \{ h_i h_i : 1\le i \le n\}[/tex]

I hope you see that these sets will probably be different. In fact the second set will only be H if the map sending x to x^2 is a bijection on H.

In general, if H and K are subgroups of G then HK is *not* a subgroup as well, by the way.
 
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  • #8
murshid_islam proved that sending x to x^2 is a surjection on H didn't he but not necessarily an injection?
 
  • #9
No. It is neither surjective, nor injective, in general.
 

Related to Group Multiplication: HH vs H - Is there a Difference?

1. What is group multiplication?

Group multiplication is a mathematical operation that combines two elements from a set to produce a third element. It is a fundamental concept in abstract algebra and is used to describe the structure of various mathematical objects, such as groups, rings, and fields.

2. What is the difference between HH and H in group multiplication?

The letters HH and H represent two different elements in a group. HH refers to the operation of multiplying two elements from the group together, while H represents a single element from the group.

3. Is there a difference between using HH and H in group multiplication?

Yes, there is a difference. When using HH, you are performing the operation of multiplication on two elements from the group. When using H, you are referring to a single element from the group.

4. Can HH and H be used interchangeably in group multiplication?

No, HH and H cannot be used interchangeably. They represent two different elements in the group, and their use will result in different outcomes.

5. How is group multiplication important in scientific research?

Group multiplication is an essential concept in abstract algebra, which is a branch of mathematics used in various scientific fields, including physics, chemistry, and computer science. It allows scientists to describe and analyze the structure and behavior of complex systems, such as molecules, particles, and networks.

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