Definition·D039
Subgroup
A subset that is itself a group under the same operation.
A subset
is a subgroup of the group
, written
, when
with
the identity and
the inverse of
in
.
In words
A subgroup is a subset containing the identity, closed under the group operation, and closed under taking inverses.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
The identity and inverses cited here are the (provably unique, L28) identity and inverses of the ambient group
. Under these conditions
really is a group in its own right: restricting
to
gives a binary operation on
by the closure clause (formally the restriction of the operation), associativity is inherited from
, and the identity and inverses are present by the other two clauses. Every group has the two trivial subgroups
and
itself (D001). Subgroups are the "sub-symmetries" of a symmetry group, and Lagrange's theorem says their sizes are severely constrained:
must divide
.
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