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Magazine Chapter 2: Problem 9
Prove that the center of a group is always a normal subgroup.
Short Answer
Expert verified
The center of a group \( Z(G) \) is a normal subgroup of \( G \).
Step by step solution
01
Recall the Definition of the Center of a Group
The center of a group, denoted as \( Z(G) \), is defined as the subset of elements of \( G \) that commute with every element of \( G \). Formally, \( Z(G) = \{ g \in G \mid gx = xg \text{ for all } x \in G \} \).
02
Recall the Definition of a Normal Subgroup
A subgroup \( N \) of a group \( G \) is called normal if for every element \( g \in G \) and \( n \in N \), the element \( gng^{-1} \) is also in \( N \). This is denoted as \( N \trianglelefteq G \).
03
Show Z(G) is a Subgroup of G
The center \( Z(G) \) is a subgroup because it is closed under the group operation (if \( a \) and \( b \) are in \( Z(G) \), then \( ab \) is also in \( Z(G) \) because \( abx = xab \) for all \( x \in G \)), contains the identity element, and includes inverses (if \( a \in Z(G) \), then \( a^{-1}x = xa^{-1} \) for all \( x \in G \)).
04
Verify Normality Condition
To prove normality, take any \( g \in G \) and \( z \in Z(G) \). We need to show \( gzg^{-1} \in Z(G) \). Since \( z \) is in the center, it commutes with any \( g \), hence \( gzg^{-1} = zg^{-1}g = ze = z \). Thus, \( gzg^{-1} = z \), an element of \( Z(G) \).
05
Conclusion
Since \( Z(G) \) satisfies both the definition of a subgroup and the normality condition, \( Z(G) \) is a normal subgroup of \( G \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Center of a Group
The term "center of a group," symbolized as \( Z(G) \), represents a fascinating concept in group theory.
Imagine a collection of elements within a group \( G \) that have a special property: they commute with every other element in \( G \).
To put it more concretely, if you pick any element \( g \) from \( G \) and another element \( c \) from the center \( Z(G) \), the equation \( gc = cg \) holds true not just sometimes, but for every single \( g \) in the group.
Imagine a collection of elements within a group \( G \) that have a special property: they commute with every other element in \( G \).
To put it more concretely, if you pick any element \( g \) from \( G \) and another element \( c \) from the center \( Z(G) \), the equation \( gc = cg \) holds true not just sometimes, but for every single \( g \) in the group.
- These elements are like universal peacemakers: no matter whom you pair them with, they always get along harmoniously.
- Mathematically, this is expressed as: \( Z(G) = \{ g \, | \, gx = xg \, \text{for all} \, x \, \text{in} \, G \} \).
- If the group were a country, \( Z(G) \) would be the council that says "We get along with everyone".
Normal Subgroup
A subgroup is a smaller group that lives inside a larger group and follows the same operation rules.
But when we talk about normal subgroups, we're discussing something even more special.
A normal subgroup \( N \) in a group \( G \) is a subgroup where, if you take any element \( g \) from \( G \) and any element \( n \) from \( N \), the combination you create in a particular sequence will still "belong" to \( N \).
But when we talk about normal subgroups, we're discussing something even more special.
A normal subgroup \( N \) in a group \( G \) is a subgroup where, if you take any element \( g \) from \( G \) and any element \( n \) from \( N \), the combination you create in a particular sequence will still "belong" to \( N \).
- This means for every \( g \) in \( G \) and every \( n \) in \( N \), the transformation \( gng^{-1} \) keeps \( n \) as a member of \( N \).
- It's like saying, "No matter how you shuffle these secret cards within the deck, they always remain unique to the club they're in."
- This stability and coherence are symbolized as \( N \trianglelefteq G \).
Group Operation
The group operation is the heart of any group.
It's how we combine elements within a group to get another element of the same group.
Think of it as a magic input-output box.
Put two members into this box, apply the operation, and you get another group member as the output.
It's how we combine elements within a group to get another element of the same group.
Think of it as a magic input-output box.
Put two members into this box, apply the operation, and you get another group member as the output.
- This operation must follow specific rules: closure, associativity, identity, and invertibility to ensure the structure we call a group.
- Closure means that any outcome of the group operation on elements from the group will yield another element within the group.
- Associativity ensures that the way in which you associate the elements when performing the operation doesn't change the eventual result.
- The identity element is a super-important element that, when combined with any other element, leaves the other element unchanged.
- Finally, invertibility means each element must have a partner that, when combined through the group operation, returns the identity element.
