• Source: Diagonal subgroup

In the mathematical discipline of group theory, for a given group G, the diagonal subgroup of the n-fold direct product G  n is the subgroup




{
(
g
,
…
,
g
)
∈

G

n


:
g
∈
G
}
.


{\displaystyle \{(g,\dots ,g)\in G^{n}:g\in G\}.}


This subgroup is isomorphic to G.




Properties and applications


If G acts on a set X, the n-fold diagonal subgroup has a natural action on the Cartesian product X n induced by the action of G on X, defined by




(

x

1


,
…
,

x

n


)
⋅
(
g
,
…
,
g
)
=
(

x

1



⋅
g
,
…
,

x

n



⋅
g
)
.


{\displaystyle (x_{1},\dots ,x_{n})\cdot (g,\dots ,g)=(x_{1}\!\cdot g,\dots ,x_{n}\!\cdot g).}


If G acts n-transitively on X, then the n-fold diagonal subgroup acts transitively on X n. More generally, for an integer k, if G acts kn-transitively on X, G acts k-transitively on X n.
Burnside's lemma can be proved using the action of the twofold diagonal subgroup.




See also


Diagonalizable group




References


Sahai, Vivek; Bist, Vikas (2003), Algebra, Alpha Science Int'l Ltd., p. 56, ISBN 9781842651575.

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