Exponent of direct product is lcm of exponents

This article gives an expression for the value of the arithmetic function exponent 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 least common multiple of the values for the direct factors.
View facts about exponent of a group: (facts closely related to exponent of a group, all facts related to exponent 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 least common multiple: (facts closely related to least common multiple, all facts related to least common multiple)

Statement

For two groups

Suppose $G_1$ and $G_2$ are groups and their exponents are $a_1,a_2$ respectively. Here, the exponent of a group is defined as the least common multiple of the orders of all elements of the group.

Then, the external direct product $G_1 \times G_2$ has exponent equal to the least common multiple of $a_1$ and $a_2$.

If a group has elements of infinite order, or if there is no finite bound on the orders of all elements, the exponent is taken to be infinite. In our case, if either $G_1$ or $G_2$ has infinite exponent, then so does $G_1 \times G_2$, whereas if both have finite exponent, then $G_1 \times G_2$ also has finite exponent given by the lcm.

For finitely many groups

Suppose $G_1,G_2,\dots,G_n$ are groups of exponents $a_1,a_2,\dots,a_n$ respectively. Then, the external direct product $G_1 \times G_2 \times \dots \times G_n$ has exponent equal to $\operatorname{lcm}(a_1,a_2,\dots,a_n)$.

In particular, if any of the groups has infinite exponent, so does the direct product, and if all the groups have finite exponent, then so does the direct product.

For internal direct product

The above statements hold if we replace external direct product by internal direct product. Note that by equivalence of internal and external direct product, the statements for both kinds of products are equivalent.

Related facts

• Exponent of semidirect product divides product of exponents, but exponent of semidirect product may be strictly less than product of exponents
• Exponent of semidirect product is a multiple of product of exponents, but exponent of semidirect product may be strictly greater than lcm of exponents