# Is the identity of a group unique?

• I
• Lebnm
In summary, the book says that if the set ##G## is a group, then every element ##g \in G## has a unique inverse ##g^{-1} \in G## such that ##g g^{-1} = g^{-1}g = e##. However, the book claims that this is written only as ##g^{-1} g = e##, and the proof given is different. First, it is required that equations ##ax=b## and ##xa=b## can be solved for x and y. Then, since ##g^{-1}g = g^{-1}e##, it follows that ##g^{-1}g = e^{-1}
Lebnm
In general, the textbooks says that, if the set ##G## is a group, so to every element ##g \in G## there is other element ##g^{-1} \in G## such that ##g g^{-1} = g^{-1}g = e##, where ##e## is the identity of the group. But I am reading a book where this propriete is write only as ##g^{-1} g = e##, and the book says that ##g g^{-1} = e## follows from this. The proof it gives is: Applaying ##g## on the left on the both sides, we have $$g(g^{-1} g) = (g g^{-1}) g = g e = g,$$and of this the book concludes that ##g g^{-1}## is equal to ##e##, because its action in ##g## gives ##g##.
Is it correct? To me, it was necessary to proof also that ##g(g g^{-1}) = g##, because with this I could use the fact that the identity of a group is unique.

To be exact, one first proves that left and right identity are the same: ##e_lg=ge_r=g \Longrightarrow e_l=e_r## and that there cannot be two identities, i.e. we have to show ##e=e'##. Then you can conclude that the right inverse is equal to the left inverse. (You wrote ##eg=ge## which you cannot know from the start, then you concluded ##gg^{-1}=e## which is also not known until the uniqueness of ##e## has been proven.)

Another way to define a group is by demanding that the equations ##ax=b## and ##xa=b## can uniquely be solved (IIRC).

Abhishek11235
A nice exercise in this area is to show that the identity of a subgroup must be the same as that of the supergroup. And that a subset S is a group iff for all a,b in S, ##ab^{-1}## is also in S.

## 1. What is the definition of group identity?

Group identity refers to the shared characteristics, beliefs, values, and behaviors that define a particular group. It is the sense of belonging and connection that individuals feel towards a group.

## 2. Why is the uniqueness of group identity important?

The uniqueness of group identity is important because it helps to differentiate one group from another and allows individuals to identify and connect with a specific group. It also helps to maintain the cohesion and solidarity within a group.

## 3. Can a group have multiple identities?

Yes, a group can have multiple identities. This can occur when a group is made up of individuals with diverse backgrounds, beliefs, and values. In such cases, the group may have different subgroups or subcultures with their own unique identities.

## 4. How is group identity formed?

Group identity is formed through a variety of factors, such as shared experiences, common goals, and similar beliefs and values. It can also be influenced by external factors, such as societal norms and cultural influences.

## 5. Can group identity change over time?

Yes, group identity can change over time. As individuals within a group evolve and grow, the group's identity may also evolve and change. External factors, such as societal changes, can also impact the identity of a group.

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