Subgroup

In group theory, a branch of mathematics, given a group [math]\displaystyle{ G }[/math] under a binary operation, a subset [math]\displaystyle{ H }[/math] of [math]\displaystyle{ G }[/math] is called a subgroup of [math]\displaystyle{ G }[/math] if the elements [math]\displaystyle{ H }[/math] also forms a group under the binary operation.

Group theory

Group theory Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product Discrete groups Classification of finite simple groups Cyclic group Zn Alternating group An Sporadic groups Mathieu group M11..12,M22..24 Conway group Co1..3 Janko group J1, 2, 3, 4 Fischer group F22..24 Baby Monster group B Monster group M Other finite groups Symmetric group, Sn Dihedral group, Dn Infinite groups The integers, Z Modular groups, PSL(2,Z) and SL(2,Z) Topological groups Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G2 F4 E6 E7 E8 Lorentz group Poincaré group Infinite dimensional groups conformal group diffeomorphism group Loop group Quantum group O(∞) SU(∞) Sp(∞)

The situation could arise in which the subset [math]\displaystyle{ H }[/math] does not comply with the binary operation. In that case, H would not be a subgroup.

For example, the even numbers are a subgroup of the integers, with addition as the binary operation.

Subgroup Media

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  Symmetric group S4

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  Alternating group 4; Cayley table; numbers

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  Dihedral group of order 8; Cayley table (element orders 1,2,2,2,2,4,4,2); subgroup of S4

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  Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4

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  Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4

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  Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,2,3,4,5)

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  Symmetric group 3; Cayley table; subgroup of S4 (elements 0,5,6,11,19,21)

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  Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,14,15,20,21)

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  Symmetric group 3; Cayley table; subgroup of S4 (elements 0,2,6,8,12,14)