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mathusers

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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 :)

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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

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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 :)

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- #2

Dick

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Did you try anything? Any ideas? Try thinking about right and left cosets.

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mathusers

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is this correct?

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Dick

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mathusers said:

is this correct?

Yep, that's pretty much it.

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mathusers

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i understand "how" to do the question but i don't understand "why" it works... thanks :)

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Dick

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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.

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.

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.

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.

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.

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