Restricted Direct Sum

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Previously, we verified in §Direct Product of Groups that arbitrary products exist in \(\Grp\), and in §Group Homomorphisms that every morphism in \(\Grp\) has an equalizer. Hence, by the argument following [Category Theory] §Limits, ⁋Example 7, \(\Grp\) is a complete category.

On the other hand, every morphism in \(\Grp\) has a coequalizer (§Group Homomorphisms, ⁋Proposition 8). Thus, if \(\Grp\) had arbitrary coproducts, it would be a cocomplete category, and therefore bicomplete.

However, as noted in §Direct Product of Groups, ⁋Lemma 1, it is not at all obvious how to endow the coproduct \(\coprod G_i\) in \(\Set\) with a group structure ([Set Theory] §Sum of Sets, ⁋Proposition 5).

Instead, we look for the answer inside the product \(\prod G_i\), whose existence is already known. By viewing each \(G_i\) as a subgroup of \(\prod G_i\) and considering the subgroup they generate together, we call the resulting group the weak direct product and verify its universal property in this post.

However, this universal property carries the additional condition that the images of homomorphisms coming from distinct \(G_i\) must commute. Consequently, the weak direct product gives the desired construction at least for abelian groups, but fails to do so for general groups. In the next post, we will show, by a different method, that a group satisfying the universal property of the coproduct exists for arbitrary groups as well.

Restricted sum

Let a family of groups \((G_i)\) and their product be given. For each \(i\), the map \(\iota_i:G_i\rightarrow\prod G_i\) sending \(g\in G_i\) to the element whose \(i\)-th component is \(g\) and whose remaining components are the identity is an injective group homomorphism; thus each \(G_i\) can be viewed as a subgroup of \(\prod G_i\) via \(\iota_i\). It is natural to ask whether the identity

\[\prod_{i\in I} G_i=\left\langle\bigcup \iota_i(G_i)\right\rangle\]

holds. This identity almost never holds when \(I\) is infinite. As the simplest example, let \(I=\mathbb{N}\) and \(G_i=\mathbb{Z}/2\mathbb{Z}=\{\bar{0}, \bar{1}\}\). Then, for instance, the left-hand side contains the element

\[(\bar{1},\bar{1},\cdots)\]

but the right-hand side contains only elements obtained by finite operations on the \(\iota_i(\bar{1})\), so it cannot contain the element above.

Definition 1 Let a family of groups \((G_i)\) be given, and fix subgroups \(H_i\) of \(G_i\). Then the subgroup consisting of those \(x\) satisfying \(\pr_ix\in H_i\) for all but finitely many \(i\) is called the restricted sum of the \(G_i\) with respect to the \(H_i\), and is denoted \(\prod^H G_i\).

In particular, when \(H_i=\{e\}\) for all \(i\), it is called the weak direct product of the \(G_i\), and is denoted simply by

\[{\prod_{i\in I}}^w G_i.\]

The notation \(\prod^H\) is not particularly good, but fortunately we are only interested in the weak direct product, so there will be no occasion to use this notation again.

By definition,

\[\left\langle\bigcup \iota_i(G_i)\right\rangle={\prod_{i\in I}}^w G_i\]

holds. In particular, since the image of \(\iota_i\) is contained in \(\prod^w G_i\), from now on we regard \(\iota_i\) as an injective homomorphism into \(\prod^w G_i\). Also, if \(I\) is finite, then the weak direct product coincides with the ordinary direct product.

Then \(\prod^wG_i\) has the following universal property.

Theorem 2 Let a family of groups \((G_i)\) and their weak direct product \(\prod^w G_i\) be given. For another group \(H\), suppose that group homomorphisms \(f_i:G_i\rightarrow H\) satisfy the condition

For any \(i\neq j\), if \(x\in G_i\) and \(y\in G_j\), then \(f_i(x)f_j(y)=f_j(y)f_i(x)\).

Then there exists a unique group homomorphism \(f:\prod^w G_i\rightarrow H\) such that \(f_i=f\circ\iota_i\) holds for every \(i\).

Proof

First we show uniqueness. If \(f, f'\) satisfy the above, then they must agree on \(\bigcup\iota_i(G_i)\), hence also on \(\prod^w G_i\), and therefore \(f=f'\).

Now we show existence. For any \(x\in \prod^w G_i\), define \(f(x)\) by the formula

\[f(x)=\prod_{i\in I} f_i(\pr_ix)\]

where \(\prod\) denotes the product of ordinary elements. Since \(x\) is an element of \(\prod^w G_i\), all but finitely many of the \(f_i(\pr_ix)\) on the right-hand side are the identity. Moreover, the non-identity factors on the right-hand side come from distinct indices, so by hypothesis they commute with each other; hence this product is well defined regardless of the order of multiplication.

That the identity \(f_i=f\circ\iota_i\) holds follows from \(f(\iota_i(g))=\prod_{j\in I}f_j(\pr_j\iota_i(g))=f_i(g)\). That \(f\) is a group homomorphism follows because, for any \(x,y\in\prod^wG_i\),

\[f(xy)=\prod_{i\in I}f_i(\pr_i(xy))=\prod_{i\in I}f_i(\pr_ix)f_i(\pr_iy)\]

holds; picking out only the finitely many indices for which at least one of \(\pr_ix\) and \(\pr_iy\) is not \(e_i\) and calling these indices \(1,\ldots, n\), this expression becomes

\[f_1(\pr_1x)f_1(\pr_1y)f_2(\pr_2x)f_2(\pr_2y)\cdots f_n(\pr_nx)f_n(\pr_ny)\]

and since \(f_i(\pr_ix)\) and \(f_j(\pr_jy)\) always commute when \(i\neq j\), this expression can be rewritten as

\[f_1(\pr_1x)f_2(\pr_2x)\cdots f_n(\pr_nx)f_1(\pr_1y)f_2(\pr_2y)\cdots f_n(\pr_ny).\]

Hence \(f(xy)=f(x)f(y)\) and \(f\) is a group homomorphism.

The condition imposed on the \(f_i\),

For any \(i\neq j\), if \(x\in G_i\) and \(y\in G_j\), then \(f_i(x)f_j(y)=f_j(y)f_i(x)\)

is one that necessarily arises, because these are exactly the conditions satisfied by the \(\iota_i\). Consequently, Theorem 2 answers our question only for abelian groups.

Using the universal property of the weak direct product, one can prove several properties analogous to those for the direct product. For instance, the following holds.

Proposition 3 Let the \(G_i\) be groups and the \(H_i\) be normal subgroups of the \(G_i\). Then the \(\prod^w H_i\) are also normal subgroups of \(\prod^w G_i\), and the quotient group is equal to \(\prod^w (G_i/H_i)\).

Proof

Consider the canonical homomorphisms \(p_i:G_i\rightarrow G_i/H_i\). For any \(x\in\prod^wG_i\), the element \(\bigl(p_i(\pr_ix)\bigr)_{i\in I}\) is the identity except for finitely many \(i\), so it is an element of \(\prod^w(G_i/H_i)\), and the map \(p:\prod^wG_i\rightarrow\prod^w(G_i/H_i)\) obtained in this way is a homomorphism since each \(p_i\) is a homomorphism.

Given an element \(y\) of \(\prod^w(G_i/H_i)\), for those \(i\) where \(\pr_iy\) is the identity we choose the representative \(e\), and for the remaining finitely many \(i\) we choose arbitrary representatives, thereby obtaining an element of \(\prod^wG_i\); hence \(p\) is surjective. Also, \(p(x)\) being the identity is equivalent to \(\pr_ix\in H_i\) for every \(i\), so \(\ker p=\prod^wH_i\). Therefore, by §Group Homomorphisms, ⁋Lemma 1, \(\prod^wH_i\) is a normal subgroup of \(\prod^wG_i\), and by §Group Homomorphisms, ⁋Theorem 2 (The first isomorphism theorem), the identity

\[\biggl({\prod_{i\in I}}^wG_i\biggr)\bigg/\biggl({\prod_{i\in I}}^wH_i\biggr)\cong{\prod_{i\in I}}^w(G_i/H_i)\]

holds.

Internal weak product

Let \(G\) be a group and \((H_i)\) a family of subgroups of \(G\). If the elements of \(H_i\) commute with the elements of \(H_j\) whenever \(i\neq j\), then there exists a homomorphism \(\iota\) from \(\prod^w H_i\) to \(G\) induced by the inclusion homomorphisms \(\iota_i:H_i\rightarrow G\). Writing the inclusion homomorphism from \(H_i\) to \(\prod^wH_i\) as \(\iota^w_i\), we have \(\iota\circ\iota^w_i=\iota_i\).

We also make the following definition.

Definition 4 In the above situation, if \(\iota\) is an isomorphism, we say that \(G\) is the internal weak direct product of the \(H_i\).

Thinking about the form of the homomorphism \(f\) constructed in Theorem 2, one can verify that \(G\) being the internal weak direct product of the \(H_i\) is equivalent to the condition

Every \(x\in G\) can be written uniquely as a product \(\prod y_i\) of a finitely supported family \((y_i)_{i\in I}\) with \(y_i\in H_i\).

If the subgroups \(H_i\) are all normal subgroups of \(G\), then \(G\) is the internal weak direct product of the \(H_i\) provided that the following additional condition is satisfied.

Proposition 5 Let \((H_i)\) be normal subgroups of a group \(G\) satisfying the two conditions

1. \(G=\bigl\langle\bigcup_{i\in I} H_i\bigr\rangle\),
2. For each \(k\), \(H_k\cap \bigl\langle\bigcup_{i\neq k} H_i\bigr\rangle=\{e\}\).

Then \(G\) is the internal weak direct product of the \(H_i\).

Proof

First, condition 2 shows in particular that \(H_i\cap H_j=\{e\}\) holds for every pair \(i\neq j\). Now choosing arbitrary \(x_i\in H_i, x_j\in H_j\),

\[x_ix_jx_i^{-1}x_j^{-1}=x_i\bigl(x_jx_i^{-1}x_j^{-1}\bigr)=\bigl(x_ix_jx_i^{-1}\bigr)x_j^{-1}\in H_i\cap H_j=\{e\}\]

shows that the elements of \(H_i\) and \(H_j\) commute. Hence the inclusion homomorphisms \(\iota_i\) induce \(\iota\) as in Theorem 2.

To show that \(G\) is the internal weak direct product of the \(H_i\), we must show that the \(\iota\) induced in this way is an isomorphism. First, by condition 1, any \(a\in G\) is obtained by finite operations on \(\bigcup H_i\). Also, since the \(H_i\) commute with each other, we can write \(a\) as

\[a=\prod_{i\in I} h_i,\qquad\text{$\supp(h_i)$ finite and $h_i\in H_i$}.\]

Setting \(h=\prod_{i\in I} \iota^w_i(h_i)\in\prod^w H_i\), we have

\[\iota(h)=\prod_{i\in I}\iota\bigl(\iota^w_i(h_i)\bigr)=\prod_{i\in I}\iota_i(h_i)=\prod_{i\in I}h_i=a\]

so \(\iota\) is surjective.

Now suppose \(\iota(a)=e\). Then we can write \(a=(a_i)_{i\in I}\) for a finitely supported family \((a_i)\) with each term in \(H_i\). From the identity

\[\iota(a)=\prod_{i\in I}\iota_i(a_i)=\prod_{i\in I} a_i=e\]

it follows that if \(\supp(a_i)\) has at least one element and \(i\in\supp(a_i)\), then

\[a_i^{-1}=\prod_{j\in I\setminus\{i\}}a_j\in H_i\cap \left\langle\bigcup_{j\neq i} H_j\right\rangle=\{e\}\]

which contradicts the assumption that \(i\in\supp(a_i)\). Hence \(\supp(a_i)\) is empty and \(a\) is the identity.

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References

[Hun] Thomas W. Hungerford, Algebra, Graduate texts in mathematics, Springer, 2003.

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