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Normal Subgroup


Let H be a subgroup of a group G. The similarity transformation of H by a fixed element x in G not in H always gives a subgroup. If

 xHx^(-1)=H

for every element x in G, then H is said to be a normal subgroup of G, written H<|G (Arfken 1985, p. 242; Scott 1987, p. 25). Normal subgroups are also known as invariant subgroups or self-conjugate subgroup (Arfken 1985, p. 242).

All subgroups of Abelian groups are normal (Arfken 1985, p. 242).


See also

Abelian Group, Group, Normal Factor, Normal Series, Quotient Group, Subgroup Explore this topic in the MathWorld classroom

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Scott, W. R. Group Theory. New York: Dover, 1987.

Referenced on Wolfram|Alpha

Normal Subgroup

Cite this as:

Weisstein, Eric W. "Normal Subgroup." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NormalSubgroup.html

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