A subgroup is a subset H of group elements of a group G that satisfies the four group requirements. It must therefore contain the identity element. "H is a subgroup of G" is written H subset= G, or sometimes H<=G (e.g., Scott 1987, p. 16).

The order of any subgroup of a group of order h must be a divisor of h.

A subgroup H of a group G that does not include the entire group G itself is known as a proper subgroup, denoted H subset G or H<G.

See also

Cartan Subgroup, Composition Series, Fitting Subgroup, Group, Normal Subgroup, Proper Subgroup Explore this topic in the MathWorld classroom

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Scott, W. R. Group Theory. New York: Dover, 1987.

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Weisstein, Eric W. "Subgroup." From MathWorld--A Wolfram Web Resource.

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