Normal subgroups – Knowledge and References

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CRPRO: the design and implementation of a test case reducer

Published in Amir Hussain, Mirjana Ivanovic, Electronics, Communications and Networks IV, 2015

A normal subgroup N of G, is a group that has no conjugates other than N itself. In other words, for every x∈N and every y∈G,y-1xy∈N. Most subgroups of matrix groups are not normal subgroups. However, if we restrict the size of G, it is possible to turn an ordinary subgroup, H, into a normal subgroup of the restricted group. The normalizer of H, written N(H), is the smallest subgroup of G in which H is normal. Since every group is a normal subgroup of itself, N(H) always exists.

Algebraic Structures

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Published in B. V. Senthil Kumar, Hemen Dutta, Discrete Mathematical Structures, 2019

B. V. Senthil Kumar, Hemen Dutta

Definition 4.2.47 Normal Subgroup:A subgroup (H, ⋆) of a group (G, ⋆) is said to be a normal subgroup of G if for every x ∈ G and for every h ∈ H, xhx−1 ∈ H or xHx−1H.

The concept of invariance in school mathematics

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Published in International Journal of Mathematical Education in Science and Technology, 2018

Shlomo Libeskind, Moshe Stupel, Victor Oxman

Higher mathematics deals with transformation of sets of objects and the identification of complete sets of invariants. Felix Klein (1849–1925) viewed geometry as a study of properties of figures that are invariant under certain transformations. In group theory, a normal subgroup N of a group G is a crucial concept. A subgroup N of a group G is normal if every left coset is also a right coset; that is if for all g ∈ G gN = Ng or equivalent gNg− 1 = N. For that reason, a normal subgroup is also called an invariant subgroup. Invariant Theory and Geometric Invariant Theory are more recent areas of abstract algebra.