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What is subgroup
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What is subgroup Related: Lecture 2 - Basic Definition and Properties...
A branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H x H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (i. e. H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (i. e. {e} ≠ H ≠ G).
If H is a subgroup of G, then G is sometimes called an overgroup of H.
The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.
This article will write ab for a ∗ b, as is usual.
This question is part of UPSC exam. View all Engineering Mathematics courses
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What is subgroup Related: Lecture 2 - Basic Definition and Properties...
A subgroup is a subset of group elements of a group that satisfies the four group requirements. It must therefore contain the identity element. " is a subgroup of " is written , or sometimes (e.g., Scott 1987, p. 16). The order of any subgroup of a group of order must be a divisor of.
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What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups
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What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups for Engineering Mathematics 2024 is part of Engineering Mathematics preparation. The Question and answers have been prepared according to the Engineering Mathematics exam syllabus. Information about What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups covers all topics & solutions for Engineering Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups.
What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups for Engineering Mathematics 2024 is part of Engineering Mathematics preparation. The Question and answers have been prepared according to the Engineering Mathematics exam syllabus. Information about What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups covers all topics & solutions for Engineering Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is subgroup Related: Lecture 2 - Basic Definition and Properties of Subgroups.
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