Proving Normality Subgroups in a Group | G/H = 2 | Step-by-Step Guide

  • Thread starter mathusers
  • Start date
In summary, the conversation discusses the concept of normal subgroups in a group. The main idea is that if a subgroup H has two left and right cosets in a group G, then it must be normal in G. This is because for any element g_1 in H, g_1H is equal to H which is also equal to Hg_1. Similarly, for any element g_2 not in H, g_2H is not equal to H or Hg_2, but since there are only two cosets, they must be equal. This proves that H is normal in G.
  • #1
mathusers
47
0
Hi next one? Any ideas here?

Let G be a group and [itex] H \subset G[/itex] a subgroup such that |G/H| = 2. Show that H is normal in G.

thnx :)
 
Physics news on Phys.org
  • #2
Did you try anything? Any ideas? Try thinking about right and left cosets.
 
  • #3
[itex][G/H]=2[/itex] means that [itex]H[/itex] has two left and right cosets in [itex]G[/itex]. Assume [itex]g_1 \in H[/itex], then it is trivial that [itex]g_1H = H = Hg_1[/itex]. Now assume [itex]g_2 \notin H[/itex], this means that [itex]g_2H \neq H \neq Hg_2[/itex]. But since there are only 2 cosets and both of them are not in H then it means they are the same so [itex]g_2H = Hg_2[/itex]

is this correct?
 
  • #4
mathusers said:
[itex][G/H]=2[/itex] means that [itex]H[/itex] has two left and right cosets in [itex]G[/itex]. Assume [itex]g_1 \in H[/itex], then it is trivial that [itex]g_1H = H = Hg_1[/itex]. Now assume [itex]g_2 \notin H[/itex], this means that [itex]g_2H \neq H \neq Hg_2[/itex]. But since there are only 2 cosets and both of them are not in H then it means they are the same so [itex]g_2H = Hg_2[/itex]

is this correct?

Yep, that's pretty much it.
 
  • #5
just a question here though, when it says [itex]g_1H = H = Hg_1[/itex], is it referring to g_1 as the identity element of the group? if so, can you please explain why this proves that H is normal in G?
i understand "how" to do the question but i don't understand "why" it works... thanks :)
 
  • #6
There are two left cosets g1H and g2H. One of them is eH=H. If g1H=H, then g1 could be e, it could also be anything else in H. As you said. g2 can be anything not in H. If you really want to understand it prove these statements.
 

FAQ: Proving Normality Subgroups in a Group | G/H = 2 | Step-by-Step Guide

1. What is a normal subgroup in a group?

A normal subgroup in a group is a subgroup that is invariant under conjugation by any element in the group. In other words, if you take an element from the group and use it to conjugate the elements in the subgroup, the resulting subgroup will still be the same. This is an important concept in group theory and is denoted by the symbol "N" or "G/H", where H is the normal subgroup.

2. How do you prove that a subgroup is normal in a group?

To prove that a subgroup is normal in a group, you can use the definition of a normal subgroup and show that it is invariant under conjugation. You can also use other methods such as the quotient group test or the index test. These methods involve showing that the quotient group (G/H) has certain properties that are only true for normal subgroups.

3. What is the significance of G/H = 2 in this context?

The notation G/H = 2 signifies that the index of the subgroup H in the group G is equal to 2. This means that the subgroup H divides the group G into two distinct cosets, which are sets of elements in G that are related to each other through the subgroup H. This information is useful in proving that the subgroup is normal in the group.

4. Can you provide a step-by-step guide for proving normality subgroups in a group?

Yes, the following are the steps for proving normality subgroups in a group:

  • Step 1: Understand the definition of a normal subgroup and the properties it must satisfy.
  • Step 2: Use the definition to show that the subgroup is invariant under conjugation by any element in the group.
  • Step 3: Use other methods such as the quotient group test or the index test to show that the quotient group (G/H) has certain properties that are only true for normal subgroups.
  • Step 4: If using the quotient group test, construct the cosets of the subgroup H and show that they form a group under the operation of the larger group G.
  • Step 5: If using the index test, show that the index of the subgroup H in the group G is equal to the number of cosets, which is 2 in this case.
  • Step 6: Conclude that the subgroup H is normal in the group G.

5. Why is proving normality subgroups important in group theory?

Proving normality subgroups is important in group theory because it helps to classify and understand different types of groups. Normal subgroups have several important properties that make them useful in studying the structure of a group. They also play a key role in the classification of groups as simple or non-simple. Additionally, normal subgroups are used in many applications, such as in cryptography and the study of symmetry in mathematics and physics.

Similar threads

Replies
9
Views
1K
Replies
7
Views
1K
Replies
5
Views
1K
Replies
7
Views
2K
Replies
1
Views
1K
Back
Top