Order of direct product is product of orders

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This article gives an expression for the value of the arithmetic function order of a group of an external direct product in terms of the values for the direct factors. It says that the value for the direct product is the product of the values for the direct factors.
View facts about order of a group: (facts closely related to order of a group, all facts related to order of a group)
View facts about external direct product: (facts closely related to external direct product, all facts related to external direct product)
View facts about product: (facts closely related to product, all facts related to product)

Statement

For two groups

Suppose G_1 and G_2 are groups of orders a_1,a_2 respectively. Then, the external direct product G_1 \times G_2 has order equal to the product a_1a_2.

When G_1 and G_2 are finite groups, so is G_1 \times G_2 and all the orders are natural numbers. When either of G_1 and G_2 is an infinite group, G_1 \times G_2 is also infinite, and the statement is interpreted in terms of infinite cardinals.

For finitely many groups

Suppose G_1,G_2,\dots,G_n are groups of orders a_1,a_2,\dots,a_n respectively. Then, the external direct product G_1 \times G_2 \times \dots \times G_n has order equal to the product a_1a_2 \dots a_n or \prod_{i=1}^n a_i.

When all the groups G_i are finite groups, so is the direct product, and all the orders are natural numbers.

When any of the groups is infinite, so is the direct product, and the statement is interpreted in terms of infinite cardinals. Note that, in this latter case, assuming the axiom of choice, the product of the infinite cardinals is the maximum of the infinite cardinals being multiplied.

For internal direct products

The statement above holds if we replace external direct product by Internal direct product (?), i.e., the order of the internal direct product is the product of the orders. Note that by equivalence of internal and external direct product, the statements for both kinds of products are equivalent.

For infinitely many groups using unrestricted direct products

Suppose I is an indexing set and G_i, i \in I is a collection of groups. Then, the order of the (unrestricted) external direct product of the G_is is the (possibly infinite) product of the orders of the G_is. Note that this infinite multiplication is done according to various rules for cardinal multiplication. The rule for computing the product is as follows: it is the maximum of the following two things:

  • The maximum of the cardinals being multiplied
  • The power cardinal of (the number of cardinals strictly greater than 1 that are being multiplied)

Note, in particular, that if each of the groups is finite, then the order of the unrestricted direct product is the power cardinal of the number of groups that have order strictly greater than 1, or equivalently, the number of nontrivial factors of the product.

For infinitely many groups using restricted direct products

Note that for infinitely many groups, the internal direct product matches, not the unrestricted external direct product, but the restricted external direct product. There is a notion of multiplication that, when fed with the orders of the groups, gives the order of the restricted external direct product. The product is defined as the maximum of the following two things:

  • The maximum of the cardinals being multiplied
  • The number of cardinals strictly greater than 1 that are being multiplied

Note, in particular, that if each of the groups is finite, then the order of the unrestricted direct product is the number of groups that have order strictly greater than 1.

Related facts

  • Lagrange's theorem states that the order of a group equals the product of the order of a subgroup and the size of its coset space. Interpreted for an internal direct product, we can take either of the direct factors as the subgroup, in which case the quotient will be isomorphic to the other direct factor. Direct products are thus a special illustration of Lagrange's theorem. However, whereas Lagrange's theorem holds only for groups and not for loops or monoids, the order of a direct product is the product of the orders in all these cases as well.

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