Subgroup: Difference between revisions

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imported>Richard Pinch
(reworded to draw distinction between classes and specific subgroups)
imported>Richard Pinch
(swapping two)
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Particular classes of subgroups include:
Particular classes of subgroups include:
{{r|Centre of a group}}
{{r|Characteristic subgroup}}
{{r|Characteristic subgroup}}
{{r|Essential subgroup}}
{{r|Essential subgroup}}
{{r|Normal subgroup}}


Specific subgroups on a given group include:
Specific subgroups on a given group include:
{{r|Centre of a group}}
{{r|Commutator subgroup}}
{{r|Commutator subgroup}}
{{r|Frattini subgroup}}
{{r|Frattini subgroup}}
{{r|Normal subgroup}}


==References==
==References==
* {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=7-8 }}
* {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=7-8 }}

Revision as of 01:32, 7 November 2008

In group theory, a subgroup of a group is a subset which is itself a group with respect to the same operations.

Formally, a subset S of a group G is a subgroup if it satisfies the following conditions:

  • The identity element of G is an element of S;
  • S is closed under taking inverses, that is, ;
  • S is closed under the group operation, that is, .

These correspond to the conditions on a group, with the exception that the associative property is necessarily inherited.

It is possible to replace these by the single closure property that S is non-empty and .

The group itself and the set consisting of the identity element are always subgroups.

Particular classes of subgroups include:

  • Characteristic subgroup [r]: A subgroup which is mapped to itself by any automorphism of the whole group. [e]
  • Essential subgroup [r]: A subgroup of a group which has non-trivial intersection with every other non-trivial subgroup. [e]
  • Normal subgroup [r]: Subgroup N of a group G where every expression g-1ng is in N for every g in G and every n in N. [e]


Specific subgroups on a given group include:

References

  • Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 7-8.