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The idempotent element in a group are
- a)inverse elements of a group
- b)identity elements of a group
- c)Any elements of a group
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The idempotent element in a group area)inverse elements of a groupb)id...
Every group has exactly one idempotent element : the identity.
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The idempotent element in a group area)inverse elements of a groupb)id...
Explanation:
In group theory, an idempotent element is an element that, when operated upon by itself, remains unchanged. In other words, if we multiply an idempotent element by itself, the result is the same element.
Identity Elements:
The identity element of a group is an element that, when operated upon any other element of the group, leaves the other element unchanged. In other words, if we multiply any element of the group with the identity element, the result is the same element.
Relation between Idempotent and Identity Elements:
An idempotent element is a special case of an identity element. If an element is idempotent, then it is also an identity element. This is because when we multiply an idempotent element by itself, the result is the same element, which satisfies the definition of an identity element.
Example:
Let's consider a group G with an idempotent element e. Now, if we multiply e with any other element a in the group, we get ae = a. But since e is idempotent, we also have ee = e. Therefore, e satisfies the condition of an identity element.
Answer:
Therefore, the correct answer is option 'B' - Identity elements of a group. Idempotent elements are a subset of identity elements in a group.
In group theory, an idempotent element is an element that, when operated upon by itself, remains unchanged. In other words, if we multiply an idempotent element by itself, the result is the same element.
Identity Elements:
The identity element of a group is an element that, when operated upon any other element of the group, leaves the other element unchanged. In other words, if we multiply any element of the group with the identity element, the result is the same element.
Relation between Idempotent and Identity Elements:
An idempotent element is a special case of an identity element. If an element is idempotent, then it is also an identity element. This is because when we multiply an idempotent element by itself, the result is the same element, which satisfies the definition of an identity element.
Example:
Let's consider a group G with an idempotent element e. Now, if we multiply e with any other element a in the group, we get ae = a. But since e is idempotent, we also have ee = e. Therefore, e satisfies the condition of an identity element.
Answer:
Therefore, the correct answer is option 'B' - Identity elements of a group. Idempotent elements are a subset of identity elements in a group.
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