A normal subgroup H of a group G is a subgroup that satisfies the condition of similarity transformation with any chosen element in G. If G is an abelian group and x is an element of G, then Hx is a right coset of H in G and xH is a left coset of G. If G is abelian then xH = Hx. However, even if G is not an abelian group, it can still satisfy xH = Hx. In this scenario, the subgroups of G are termed as normal subgroups, invariant subgroups or distinguished subgroups.
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A subgroup H of a group G is considered a normal subgroup if for every x in G and for h in H, xh = h, that is, xhx-1 belongs to H. Since the above statement holds true for all h in H, we can define normal subgroups of a group G as:
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A subgroup H of a group G is a normal subgroup if and only if xHx-1 ⊆ H for every x in G, where x may or may not be in H. |
Normal subgroups are sometimes also referred to as self-conjugates. Normal subgroups are denoted as H ◁ G, it is read as “H is a normal subgroup of G”.
Every group has necessarily two trivial normal subgroups, viz., the single identity element of G and G itself.
geg-1 = gg-1 = e ∈ {e}
Thus {e} is the normal subgroup of G.
ghg-1 = k which could be any element in G. Thus, G itself is a normal subgroup.
Example 1:
Prove that a subgroup H of a group G is a normal subgroup if and only if g-1Hg = H for every g in G.
Solution:
We first take for H be a subgroup of H, g-1Hg = H for every g in G. Then
gHg-1 = H for every g in G
⇒ gHg-1 ⊆ H for every g in G
Hence, by the definition of normal subgroup H is a normal subgroup of G.
Example 2:
Prove that all abelian groups have normal subgroups.
Solution:
Let G be an abelian group and H be a subgroup of G. Since G is abelian therefore all elements of G commutative with respect to multiplication. Let g ∈ G and h ∈ H as H is a subgroup of G the h must belong to G also, then
hg = gh for all g in G
⇒ ghg-1 = h ∈ H for all g in G
⇒ gHg-1 ⊆ H for all g in G
Hence, every abelian group has normal subgroups.
What is a normal subgroup in group theory?
A normal subgroup of group G consists of all those elements which remain invariant by conjugation of all elements of G. That is, if H be a subgroup of G and for h in H, ghg-1 = h for every g in G, then H is called a normal subgroup of G.
What are the different names of a normal subgroup?
Normal subgroups are also known as Self-conjugate, invariant, distinguished subgroups.
How many normal subgroups does a group have?
Every group has two trivial normal subgroups which are The subgroup that has only the identity element of G and another is G itself. For an abelian group every subgroup is normal.
Do non-abelian groups have normal subgroups?
Yes, every group whether abelian or non-abelian, have at least two trivial normal subgroups, the identity element and the group itself. A group which is not abelian but have a normal subgroup other than the trivial is called the Hamiltonian group.