Problem 35

Question

Let $G$ be a group, $a \in G$. Show that the centralizer $C(a)=G$ if and only if $a \in$ $Z(G),$ the center of $G$

Step-by-Step Solution

Verified

Answer

$C(a) = G$ if and only if $a \in Z(G)$.

1Step 1: Understanding the Problem

The problem is asking us to show that the centralizer of an element $a$ in a group $G$ is the entire group $G$ if and only if $a$ is in the center of the group, $Z(G)$. The centralizer $C(a)$ consists of elements in $G$ that commute with $a$, while the center $Z(G)$ consists of elements in $G$ that commute with every element of $G$.

2Step 1: Define Centralizer and Center

The centralizer of $a$, denoted $C(a)$, is defined as $\{g \in G \mid ga = ag\}$. The center of the group $G$, denoted $Z(G)$, is defined as $\{g \in G \mid gx = xg \text{ for all } x \in G\}$.

3Step 2: Show if $C(a) = G$ then $a \in Z(G)$

Assume $C(a) = G$, meaning every element $g \in G$ commutes with $a$. Thus, $ga = ag$ for every $g$ in $G$. By definition, this means $a$ commutes with every element of $G$, so $a$ belongs to $Z(G)$.

4Step 3: Show if $a \in Z(G)$ then $C(a) = G$

Now assume $a \in Z(G)$. Then, $a$ commutes with every $g \in G$, so $ga = ag$ for all $g$. This means every element of $G$ is in $C(a)$, and hence $C(a) = G$.

5Step 4: Conclusion

Both directions have been proven: If the centralizer of $a$ is $G$, then $a$ is in the center $Z(G)$, and if $a$ is in $Z(G)$, then the centralizer of $a$ is $G$. Therefore, $C(a) = G$ if and only if $a \in Z(G)$.

Key Concepts

Centralizer in GroupCenter of GroupCommuting Elements

Centralizer in Group

A centralizer in group theory is an important concept that refers to a collection of elements within a group that commute with a particular element. Let's explore this further.The centralizer of an element $a$ in a group $G$, usually denoted as $C(a)$, is defined as the set of all elements $g$ in $G$ for which the equation $ga = ag$ holds true. This simply means that when you multiply any element $g$ from this set with $a$, you will get the same result as if you multiplied $a$ with $g$. They essentially ``get along" well.

Formula: $C(a) = \{g \in G \mid ga = ag\}$
Elements in $C(a)$ have this commutative property specific to the element $a$.

Understanding the centralizer helps us understand how certain elements relate to others within the group. When we say $C(a) = G$, this indicates every element in $G$ commutes with $a$. This can have significant implications for the structure of the group as a whole.

Center of Group

The center of a group is another fundamental concept in group theory. It is crucial for understanding the symmetry and structure of a group.The center of a group $G$, denoted as $Z(G)$, refers to the subset of $G$ consisting of elements that commute with every other element in $G$. In other words, if an element is in the center, it does not matter in which order it is combined with any other element; the result will be the same.

Formula: $Z(G) = \{g \in G \mid gx = xg \text{ for all } x \in G\}$
Elements in $Z(G)$ are considered to be the most 'neutral' since they do not disturb the arrangement of other elements.

If an element $a$ is part of $Z(G)$, it means $a$ is perfectly harmonious with all elements of the group. Consequently, if an element's centralizer is the whole group $G$, it implies that this element itself is in the center, demonstrating an essential symmetry characteristic.

Commuting Elements

Commuting elements in a group are those which can be operated on each other in any order, and the result will remain unchanged.In the context of group theory, two elements $a$ and $b$ in a group $G$ commute if their product is the same regardless of the order: $ab = ba$. This property is fundamental in discussions related to the centralizer and the center of a group.

Commutativity property: $ab = ba$
For each element $a$ in a group, all elements $b$ that commute with $a$ form the centralizer $C(a)$.

Commutativity is a vital concept because it helps define more structured subsets within a group. In many algebraic structures, finding elements that commute could simplify complex computations. When you understand which elements commute, you gain insight into the group's internal organization and its potential symmetries.

Problem 34

Problem 36

Other exercises in this chapter

Problem 33

Show that if $\sigma \in A_{n}$, then $\sigma$ can be written as a product of 3 -cycles.

View solution

Problem 34

Show that if $\sigma \in A_{n}$, then $\sigma$ can be written as a product of 3 -cycles $(12 s)$ where $s=3,4, \ldots, n$

View solution

Problem 36

Find the center $Z\left(S_{3}\right)$ of $S_{3}$.

View solution

Problem 37

Show that if $m \leq n,$ then the number of $m-$ cycles $\left(a_{1}, a_{2}, \ldots, a_{m}\right)$ in $S_{n}$ is $$ n(n-1)(n-2) \ldots(n-m+1) / m $$

View solution