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Let $G,H$ be groups. Suppose we have both an action of $G$ on $H$, and an action of $H$ on $G$, both non-trivial. Let "$\cdot$" define the former action, and $\circ$ define the latter.

What can we say about the product: $$ (g,h)\,(g',h') := \big(g\,(h\circ g'), h\,(g\cdot h')\big), $$

or the product: $$ (g,h)\,(g',h') := \big(g\,(h\circ g'), (g'\cdot h)\,h'\big), $$

or anything similar where the other action is a right action?

Do we have a group? Is this used somewhere? Is it useful, or too general?

Thank you.


marked as duplicate by user1729, user63181, Namaste, naslundx, 6005 Apr 29 '14 at 12:40

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    $\begingroup$ You might compare to en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p_product $\endgroup$ – Jack Schmidt Apr 28 '14 at 20:00
  • $\begingroup$ Probably this operation fails to be associative unless you specify a compatibility between the two actions (which should just be the last two axioms in the Wikipedia article Jack Schmidt linked to). $\endgroup$ – Qiaochu Yuan Apr 28 '14 at 22:41