2

Cancellation theorem in group theory (for direct product) says that if $B$ is a finite group and $A \times B \simeq A_1 \times B_1$ and $B \simeq B_1$ then $A \simeq A_1.$

Of course, if $B$ is not finite, the result is absurd, even for finitely presented groups (Here is an example by Steve)

I wonder whether the cancellation theorem holds for different products (in finite or infinite cases), such as semi-direct product, free product, fiber product over a given group, Zappa-Szep product (knit product), Wreath product.

YCor
  • 60,149
Ehsan M. Kermani
  • 1,651
  • 1
    Which of these various examples have you already tried? – Yemon Choi Dec 14 '11 at 05:58
  • 1
    There's an inherent lack of symmetry in other products, like the semidirect product or wreath product. Which of the two subgroups involved would you like to cancel? I'm pretty sure there are counterexamples for either one though. – Steve D Dec 14 '11 at 06:03
  • Also, from the Grushko decomposition theorem, it would seem that for free products, cancellation is possible for finitely presented groups. – Steve D Dec 14 '11 at 06:12
  • @Yemon Choi: I tried for the semi-direct product of finite groups, but nothing came out! – Ehsan M. Kermani Dec 14 '11 at 06:25
  • @Steve D: That's a good point! I mean, for each sides of the semi-product, and wreath product (which I'm not very familiar and comfortable with the latter one) – Ehsan M. Kermani Dec 14 '11 at 06:31
  • 1
    Consider the maps of short exact sequences in the case of semidirect products... – David Roberts Dec 14 '11 at 06:43
  • 1
    Here's a counterexample for knit products: http://mathforum.org/kb/thread.jspa?forumID=13&threadID=2030472&messageID=6954692#6954692

    Of course, knit products and semidirect products are more general than direct products, so the counterexample you cited above works in both those cases as well.

     – Steve D Dec 14 '11 at 08:04
  • 3
    I think the question is a little bit to vague and unfocused. – Martin Brandenburg Dec 14 '11 at 11:44

1 Answers1

3

For a straightforward example that the subgroup doing the acting in a semidirect product cannot automatically be cancelled (and as semidirect products are knit products, also addresses that case), the dihedral group of order $8$ can be written as a semidirect product in two different ways: $C_4\rtimes C_2$ (with the action being inversion) and $(C_2\times C_2)\rtimes C_2$ (with the action being swap coordinates).

John McVey
  • 1,018
  • FWIW, the second of these is even a wreath product: $(C_2\times C_2)\rtimes C_2 = C_2\wr C_2$, though I admit this doesn't compare a wreath product to a wreath product. – John McVey Apr 19 '19 at 14:39