Talk:Complement (group theory)

Two definitions

The two stated definitions are not equivalent in general, only if one of the groups is normal. (Example: one can generate the symmetric group $S_{n}$ by a cycle and a transposition. If one lets $G=S_{n}$, $H$ be the group generated by the cycle and $K$ the group generated by the transposition, then $G\not =HK$ (for example because $HK$ has too few elements).

We're not sure what the correct definition is, we'd guess the first one, as it corresponds to the complement in the lattice, when one considers lattice of subgroups of $G$.

Matt DeVos and Robert Samal

The standard definition in finite group theory is actually the second one: H ≤ G has a complement K if G=HK and H∩K=1. The idea is that complements in the group theory sense are pretty rare. I'll see about adding some specific citations to standard sources. JackSchmidt (talk) 00:17, 6 August 2009 (UTC)

Here is a little survey:

• AlperinBell GaR: only normal
• Dixon's PiGT: page 53, definition is the second type
• DoerkHawkes's FSG: page 4, definition is the second type
• Gorenstein's FG: only normal.
• Hall's ToG: only p-complement
• Huppert's EG: only normal
• Robinson's C in ToG: page 246, definition is the second type
• Rose's C on GT: page 251, definition is the second type
• Rotman's ToG: page 167, definition is the second type
• Scott's GT: page 137, definition is the second type
• Suzuki's GT: only normal
• Wehrfritz's SCGT: only normal

So looks like second type definition, G=HK, is the standard, though it is also fairly standard just to define "normal complement", "p-complement", and "complement of normal subgroup" without defining the general concept. JackSchmidt (talk) 01:26, 6 August 2009 (UTC)

Translated to Japanese.

Hi! I appreciate your wonderful article. I translated this article and create a new Japanese edition. The title is "補群", pronounced hogun. (Here, "補" means complement, and "群" means group.)

Sorry, some of the translation is not completed. ja:補群 How can I create a inter-languate link? (I'm a beginner Wikipedian, I don't know how to link between languages. Or, could you create the link?)

Sincerely yours, Mr T.I.71 (talk) 13:20, 29 March 2023 (UTC)