As a group owner, how can I prevent members from posting the same message to multiple groups, including mine?

Moderating members posts is the only way to prevent it.

left tabs on homepage click management- under group settings click messages- next page Posting and Archives (Edit) <-- click the edit- 3rd column Moderation check either new members OR Unmoderated - click save changes-

REMEMBER

this will only affect those who join after this setting change. members now prior to this remain on the previous setting that you had, lets say no moderation.

ALSO

these members stay moderated until you individually reset each one and here's how (this is also how you'd moderate a member who needs it)

locate said member is members list- click edit membership below their ID- Posting Messages: Override: This member's posts are always moderated (Edit) <-- click the edit

next page choose either

* Messages posted by this member are not moderated.

* Messages posted by this member are moderated.

click save changes

You could put the offenders on moderation then not approve their posts.