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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexsome minimum, for cross-linking with [[Sp(n).Sp(1)]] <a href="https://ncatlab.org/nlab/revision/central+products+of+groups/1">v1</a>, <a href="https://ncatlab.org/nlab/show/central+products+of+groups">current</a>

   some minimum, for cross-linking with Sp(n).Sp(1)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexchanged page name to singular <a href="https://ncatlab.org/nlab/revision/central+product+of+groups/1">v1</a>, <a href="https://ncatlab.org/nlab/show/central+product+of+groups">current</a>

   changed page name to singular

   v1, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 16th 2019
   • (edited May 16th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhen I started this entry, I deliberately changed the wording of the definition as compared to what one finds in most of the literature. I have added now a remark on this point, for clarity: *** Beware that most texts insists on stating the choices in the definition of the central product as that of Beware that most texts insists on stating the choices in Def. \ref{CentralProduct} as that of 1. two separate subgroups $C_i \xhookrightarrow{\iota_i} Z(G_i)$ 1. an [[isomorphism]] $C_1 \xrightarrow[\simeq]{\phi} C_2$ between them and insists that the second group acts via $(-)^{-1}\circ \phi$. These clauses matter if one thinks of the subgroup inclusions as in [[material set theory]]. But we speak [[structural set theory]], which means that a [[subgroup]] inclusion as in (eq:SubgroupInclusion) is really a choice of _[[monomorphism|monic]] [[homomorphism]]_, and this choice already absorbs the choice of $\phi$ and or $(-)^{-1}\circ \phi$ above. <a href="https://ncatlab.org/nlab/revision/diff/central+product+of+groups/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/central+product+of+groups/4">v4</a>, <a href="https://ncatlab.org/nlab/show/central+product+of+groups">current</a>

   When I started this entry, I deliberately changed the wording of the definition as compared to what one finds in most of the literature.

   I have added now a remark on this point, for clarity:

   ---

   Beware that most texts insists on stating the choices in the definition of the central product as that of

   Beware that most texts insists on stating the choices in Def. \ref{CentralProduct} as that of

   1. two separate subgroups $C_i \xhookrightarrow{\iota_i} Z(G_i)$

   2. an isomorphism $C_1 \xrightarrow[\simeq]{\phi} C_2$ between them

   and insists that the second group acts via $(-)^{-1}\circ \phi$.

   These clauses matter if one thinks of the subgroup inclusions as in material set theory. But we speak structural set theory, which means that a subgroup inclusion as in (eq:SubgroupInclusion) is really a choice of monic homomorphism, and this choice already absorbs the choice of $\phi$ and or $(-)^{-1}\circ \phi$ above.

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 21st 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded mentioning of the example of [[spin^c-groups]] <a href="https://ncatlab.org/nlab/revision/diff/central+product+of+groups/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/central+product+of+groups/5">v5</a>, <a href="https://ncatlab.org/nlab/show/central+product+of+groups">current</a>

   added mentioning of the example of spin^c-groups

   diff, v5, current