In mathematics, a topological group G is called the topological direct sum[1] of two subgroups H1 and H2 if the map
\( {\displaystyle {\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}} \)
is a topological isomorphism.
More generally, G is called the direct sum of a finite set of subgroups \( H_i, i=1,\ldots, n \) of the map
\( \begin{align} \prod^n_{i=1} H_i&\longrightarrow G \\ (h_i)_{i\in I} &\longmapsto h_1 h_2 \cdots h_n \end{align} \)
Note that if a topological group G is the topological direct sum of the family of subgroups \( H_i \) then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family \( H_i \) .
Topological direct summands
Given a topological group G, we say that a subgroup H is a topological direct summand of G (or that splits topologically from G) if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K.
A the subgroup H is a topological direct summand if and only if the extension of topological groups
\( 0 \to H\stackrel{i}{{} \to {}} G\stackrel{\pi}{{} \to {}} G/H\to 0 \)
splits, where i is the natural inclusion and \( \pi \) is the natural projection.
Examples
Suppose that G is a locally compact abelian group that contains the unit circle \( \mathbb T \) as a subgroup. Then \( \mathbb T \) is a topological direct summand of G. The same assertion is true for the real numbers \( \mathbb {R} \) [2]
References
E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR 0551496 (81k:43001)
Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. ISBN 0-8247-1507-1 MR 0637201 (83h:22010)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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