In
mathematics, given a
group G and two
subgroups H and
K of
G, one can define the
product of
H and
K, denoted by
HK as the set of all
elements of the form
hk, for all
h in
H and
k in
K. In
general
HK is not a subgroup (
hkh'k' is not of the form
hk);
it is a subgroup if and only if one among
H and
K is a
normal subgroup of
G. Indeed, if this is the case (assume
K is normal),
hkh'k' =
hh' h' ^{-1}kh'k' , and
h' ^{-1}kh is
an element of
K, so that
hh' is in
H and
h' ^{-1}kh'k' is in
K, as required. An analogous argument shows that (
hk)
^{-1} is of the form
h'k' .
Of particular interest are products enjoying further properties, the semidirect product and the direct product. They allow also to construct a product of two groups not given as subgroups of a fixed group.
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