Normal Subgroups with |H|=2 & 3: Is Z(G) Involved?

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In summary: In your attempt at a solution what is it that you're trying to show? What do the steps you've written mean in relation to what you're trying to show? In summary, the conversation discusses how to show that a normal subgroup H of group G with a cardinality of 2 is also a subgroup of the center of G (Z(G)). The conversation also considers whether this is true when H has a cardinality of 3 and how to approach this question.
  • #1
hsong9
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Homework Statement


If H is a normal subgroup of G and |H| = 2, show that H is a subgroup of Z(G).
Is this true when |H| = 3?


The Attempt at a Solution


Since H is a normal subgroup of G and |H| = 2 = {1,a},
a in Z(G), also aa = a2 = 1 in Z(G)
aa-1 = a in Z(G). Therefore H is a subgroup of Z(G).
I am not sure my approach is correct.
and I have no idea next question when |H| =3, is it false? why??
 
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  • #2
In your attempt at a solution what is it that you're trying to show? What do the steps you've written mean in relation to what you're trying to show?

In other words, what does it mean for H to be in Z(G)? Because at no point have you addressed what that is. In fact the first thing you assert is that a is in Z(G), but that is part of the result that you're trying to show.

When you correctly prove this part, it will lead you to why it is false for |H|=3.
 
  • #3
hmm..
To show that H is a subgroup of Z(G),
1Z(G) in H
whenever a,b in H then ab in H
whenever a in H then a-1 in H.

Since both H and Z(G) are normal in G, 1Z(G) is clearly in H.
Since |H| = 2, H={1,a}. So 1,a in H then a1=1a in H
also since aa = a2 = 1 = aa-1, so a = a-1 self-inverse.

When |H|=3, we can't determine whether a2 in Z(G) or not.

how about this approach??
 
  • #4
I'll ask again: what is the definition of Z(G)? You have to show that H is a subgroup of Z(G), so at some point using the definition of Z(G) will be necessary. You have assumed that H is a subset of Z(G) without any justification for this.

What you have shown is that if H<G, and H is a subset of K for K<G, then H<K. Well, that is trivially true and immaterial.
 
  • #5
hsong9 said:

Homework Statement


If H is a normal subgroup of G and |H| = 2, show that H is a subgroup of Z(G).
Is this true when |H| = 3?


The Attempt at a Solution


Since H is a normal subgroup of G and |H| = 2 = {1,a},
a in Z(G), also aa = a2 = 1 in Z(G)
Here is the crucial point: you start by asserting that a is in Z(G). How do you know that is true? As matt grime said, you have not used the definition of Z(G). You have also not used the fact that H is a normal subgroup of G.

aa-1 = a in Z(G). Therefore H is a subgroup of Z(G).
I am not sure my approach is correct.
and I have no idea next question when |H| =3, is it false? why??
 

1. What is a normal subgroup?

A normal subgroup is a subset of a group that is invariant under conjugation by any element of the group. This means that for any element in the normal subgroup, when conjugated with any element in the original group, the result is still in the normal subgroup.

2. What does |H|=2 & 3 mean in this context?

The notation |H| is used to represent the size or order of a subgroup. In this case, |H|=2 & 3 means that the subgroup H has two or three elements.

3. How is Z(G) involved with normal subgroups of size 2 and 3?

Z(G), also known as the center of a group G, is a subset of G that contains all elements that commute with every element in G. In the case of normal subgroups with |H|=2 & 3, Z(G) is involved because it is a normal subgroup itself and has a specific relationship with the other normal subgroups of size 2 and 3.

4. What is the significance of normal subgroups with |H|=2 & 3?

Normal subgroups with |H|=2 & 3 have special properties and are important in the study of group theory. They can help to determine the structure and properties of a larger group and are useful in proving theorems and solving problems related to group theory.

5. How are normal subgroups with |H|=2 & 3 used in practical applications?

Normal subgroups with |H|=2 & 3 have applications in various areas of mathematics, including number theory, abstract algebra, and cryptography. They also have important implications in physics, particularly in the study of symmetries and conservation laws.

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