# Exponent of the direct sum of finite groups, specifically, $\sum^t_i S_{N_i} \wr D_{m_i}$

I have one general and one specific questions.

1. What is the expression for the exponent of the direct sum of finite groups?
2. What is the exponent of $\sum^t_i S_{N_i} \wr D_{m_i}$?

Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of order $2 m_i$.

My effort: I guess the exponent should be the lcm of the exponent of each $S_{N_i} \wr D_{m_i}$.

Am I correct?

• Are you sure you mean direct sum? That is usually synonymous with coproduct, which will almost always have infinite exponent. – Tobias Kildetoft Dec 18 '15 at 16:59
• Maybe it helped if you wrote what you mean by direct sum. That term is usually only used for abelian groups, and direct product is used in general. When there are an infinite number of terms, these things differ for abelian groups, but the thing for general groups that behaves like the direct sum is usually called the restricted direct product, rather than the direct sum. – Tobias Kildetoft Dec 18 '15 at 19:35
• Right, that is the definition of direct product, which is also called direct sum for abelian groups (and sometimes, but not so often, used for nonabelian groups). – Tobias Kildetoft Dec 18 '15 at 19:45
• Yes, for a finite number of terms, this will always be the case. – Tobias Kildetoft Dec 18 '15 at 19:48
• BTW, the exponent of a direct sum of a finite number of group is just the lcm of the individual exponents, so in this case it is just the exponent of the group you started with. – Tobias Kildetoft Dec 18 '15 at 19:57