What is the group action of G on itself by left conjugation?

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In summary, the textbook defines an action of a group ##G## on itself by left conjugation, where each element ##g## maps ##G## to itself through the function ##x \mapsto gxg^{-1}##. This function is also known as an inner automorphism or "conjugation by ##g##". This can also be described as a group homomorphism from ##G## to the group of inner automorphisms, ##Inn(G)##, which is a subgroup of the automorphism group ##Aut(G)##. The notation "acts / operates on" with a dot is equivalent to the introduction of this homomorphism ##\varphi## and replacing the dot with ##\varphi
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Mr Davis 97
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My textbook says the following: "Let ##G## be a group and ##G## act on itself by left conjugation, so each ##g \in G## maps ##G## to ##G## by ##x \mapsto gxg^{-1}##". I am confused by the wording of this. ##g## itself is not a function, so how does it map anything at all? I am assuming this is supposed to mean that ##g \cdot x = gxg^{-1}## is the definition of the action, but why does it use the map notation as if ##g## is a function, when ##g## is really an element?
 
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Mr Davis 97 said:
My textbook says the following: "Let ##G## be a group and ##G## act on itself by left conjugation, so each ##g \in G## maps ##G## to ##G## by ##x \mapsto gxg^{-1}##". I am confused by the wording of this. ##g## itself is not a function, so how does it map anything at all? I am assuming this is supposed to mean that ##g \cdot x = gxg^{-1}## is the definition of the action, but why does it use the map notation as if ##g## is a function, when ##g## is really an element?
It's ##\varphi : G \longrightarrow Inn(G)## with ##g \longmapsto \iota_g## where ##\iota_g : x \longmapsto gxg^{-1}## is the inner automorphism "conjugation by ##g##". ##\varphi## is a group homomorphism from ##G## to the group of inner automorphisms which is a subgroup of the automophism group ##Aut(G)## of ##G##. Whether you say "##G## acts on ##X##" or ##G \longmapsto Aut(X)## is a group homomorphism is the same thing. The notation "acts / operates on" with a dot is only a bit shorter than to introduce this homomorphism ##\varphi## and replace the dot by ##\varphi##.
 
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FAQ: What is the group action of G on itself by left conjugation?

What is the definition of "wording of a group action"?

The wording of a group action refers to the specific language or phrasing used to describe a collective or collaborative effort taken by a group of individuals. It can also refer to the way in which instructions or guidelines are written for a group to follow in order to achieve a common goal.

Why is the wording of a group action important?

The wording of a group action is important because it can impact how well a group works together and how successful they are in achieving their goals. Clear and concise wording can help ensure that all members of the group are on the same page and understand their roles and responsibilities.

How should the wording of a group action be determined?

The wording of a group action should be determined through open communication and collaboration among all members of the group. It is important to consider the perspectives and ideas of everyone involved and to strive for clarity and inclusivity in the wording.

What are some common challenges when deciding on the wording of a group action?

Some common challenges when deciding on the wording of a group action include differences in opinions and communication styles among group members, as well as the potential for misinterpretation or confusion. It is important to address these challenges and work towards finding a consensus for the best wording.

How can the wording of a group action be improved?

The wording of a group action can be improved by actively seeking and considering feedback from group members, using clear and concise language, and revising as needed. It can also be helpful to have a designated facilitator or mediator to help guide the process and ensure that all voices are heard.

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