Is H a Normal Subgroup If Every Left Coset Equals a Right Coset?

Thread starter nbacombi
Start date Dec 7, 2010
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Normal Normal subgroup Subgroup

In summary, a normal subgroup is a subgroup of a group where the left and right cosets are the same. To prove that H is a normal subgroup, you must show that for any element h in H and any element g in the larger group, the element g⁻¹hg is also in H. The significance of H being a normal subgroup is that it allows for the formation of quotient groups and has useful properties in group theory. A subgroup cannot be both normal and non-normal, and proving H is a normal subgroup affects the larger group by providing a normal subgroup structure and insight into the group's properties.

Dec 7, 2010

nbacombi

Let G be a group and H be a subgroup of G. If every left coset xH, where x in G, is equal to a right coset Hy, for some y in G, prove H is normal subgroup.

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Dec 7, 2010

lanedance

Homework Helper

so what is the definition of normal subgroup?

FAQ: Is H a Normal Subgroup If Every Left Coset Equals a Right Coset?

What does it mean for H to be a normal subgroup?

A normal subgroup is a subgroup of a group where the left and right cosets are the same. In other words, the subgroup is invariant under conjugation by elements of the larger group.

How do you prove that H is a normal subgroup?

To prove that H is a normal subgroup, you must show that for any element h in H and any element g in the larger group, the element g⁻¹hg is also in H. This can be done using the definition of a normal subgroup or by showing that the left and right cosets are equal.

What is the significance of H being a normal subgroup?

A normal subgroup is significant because it allows for the formation of quotient groups, which are important in many areas of mathematics. Additionally, normal subgroups have several useful properties that make them easier to work with in group theory.

Can a subgroup be both normal and non-normal?

No, a subgroup cannot be both normal and non-normal. A subgroup is either normal or it is not. In other words, normality is a binary property and cannot be both true and false at the same time.

How does proving H is a normal subgroup affect the larger group?

If H is proven to be a normal subgroup, then the larger group is said to have a normal subgroup structure. This means that the group can be decomposed into cosets of the normal subgroup, which can provide insight into the structure and properties of the larger group.