International Journal For Multidisciplinary Research

E-ISSN: 2582-2160   •   Impact Factor: 9.24

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Foundations Of Subgroups And The Subgroup Criterion

Author(s) Mr. M KRISHNA MOHAN, Mr. B SHIVA KUMAR
Country India
Abstract This research paper, titled "Foundations of Sub groups and the Subgroup Criterion, "offers a detailed foundational study of one of the most critical structural components in abstract algebra: the subgroup. A subgroup is defined as a subset of a group that maintains the group structure under the inherited binary operation. Understanding these internal structures is essential for classifying groups and proving key theorems.
The paper begins by reviewing the four group axioms—closure, associativity, identity, and inverse—before focusing on the core problem: establishing if a subset is a subgroup without checking all four axioms directly. This leads to the central topic: the Subgroup Criterion. We meticulously present and prove the efficiency of the one-step test (for finite groups) and the two-step test (for general groups), which dramatically simplifies the verification process.
Illustrative examples are provided, analyzing subsets of both commutative groups, such as the additive group of integers (Z, +), and non-commutative groups, such as the symmetric group S3. Ultimately, this paper formalizes the methodology for discovering the internal architecture of any given group, providing a necessary prerequisite for exploring advanced concepts like cosets, normal subgroups, and homomorphic mappings
Keywords Subgroups, Subgroup Criterion, Group Theory, Cyclic Groups, Subset Testing, Normal Subgroups, Cosets, Group Homomorphisms
Field Mathematics
Published In Volume 7, Issue 6, November-December 2025
Published On 2025-12-25
DOI https://doi.org/10.36948/ijfmr.2025.v07i06.64543
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E-ISSN 2582-2160
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