Difference between revisions of "Centralizer"
From Online Dictionary of Crystallography
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+ | The '''centralizer''' ''C<sub>G</sub>(g)'' of an element ''g'' of a group ''G'' is the set of elements of ''G'' which commute with ''g'': | ||
+ | : ''C<sub>G</sub>(g)'' = {''x'' ∈ ''G : xg = gx''}. | ||
+ | If ''H'' is a [[subgroup]] of ''G'', then ''C<sub>H</sub>(g)'' = ''C<sub>G</sub>(g) ∩ H''. | ||
− | + | More generally, if ''S'' is any subset of ''G'' (not necessarily a subgroup), the centralizer of ''S'' in ''G'' is defined as | |
− | : C<sub>G</sub>( | + | : ''C<sub>G</sub>(S)'' = {''x'' ∈ ''G'' : ∀ ''s'' ∈ ''S'', ''xs = sx''}. |
− | If | + | If ''S'' = {''g''}, then ''C(S) = C(g)''. |
− | + | ''C(S)'' is a subgroup of ''G''; in fact, if ''x, y'' are in ''C(S)'', then ''xy''<sup> −1</sup>''s'' = ''xsy''<sup>−1</sup> = ''sxy''<sup>−1</sup>. | |
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− | |||
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− | C(S) is a subgroup of G; in fact, if x, y are in C(S), then ''xy''<sup> −1</sup>''s'' = xsy<sup>−1</sup> = sxy<sup>−1</sup>. | ||
==Example== | ==Example== | ||
− | * The set of symmetry operations of the point group 4''mm'' which commute with 4<sup>1</sup> is {1, 2, 4<sup>1</sup> and 4<sup> | + | * The set of symmetry operations of the point group 4''mm'' which commute with 4<sup>1</sup> is {1, 2, 4<sup>1</sup> and 4<sup>−1</sup>}. The centralizer of the fourfold positive rotation with respect to the point group 4''mm'' is the subgroup 4: ''C''<sub>4''mm''</sub>(4) = 4. |
− | * The set of symmetry operations of the point group 4''mm'' which commute with m<sub>[100]</sub> is {1, 2, m<sub>[100]</sub> and m<sub>[010]</sub>}. The centralizer of the m<sub>[100]</sub> reflection with respect to the point group 4''mm'' is the subgroup | + | * The set of symmetry operations of the point group 4''mm'' which commute with ''m''<sub>[100]</sub> is {1, 2, ''m''<sub>[100]</sub> and ''m''<sub>[010]</sub>}. The centralizer of the ''m''<sub>[100]</sub> reflection with respect to the point group 4''mm'' is the subgroup ''mm''2 obtained by taking the two mirror reflections normal to the tetragonal '''a''' and '''b''' axes: ''C''<sub>4''mm''</sub>(''m''<sub>[100]</sub>) = ''mm''2. |
Revision as of 10:27, 13 May 2017
Centralisateur (Fr). Zentralisator (Ge). Centralizzatore (It). 中心化群 (Ja).
The centralizer C_{G}(g) of an element g of a group G is the set of elements of G which commute with g:
- C_{G}(g) = {x ∈ G : xg = gx}.
If H is a subgroup of G, then C_{H}(g) = C_{G}(g) ∩ H.
More generally, if S is any subset of G (not necessarily a subgroup), the centralizer of S in G is defined as
- C_{G}(S) = {x ∈ G : ∀ s ∈ S, xs = sx}.
If S = {g}, then C(S) = C(g).
C(S) is a subgroup of G; in fact, if x, y are in C(S), then xy^{ −1}s = xsy^{−1} = sxy^{−1}.
Example
- The set of symmetry operations of the point group 4mm which commute with 4^{1} is {1, 2, 4^{1} and 4^{−1}}. The centralizer of the fourfold positive rotation with respect to the point group 4mm is the subgroup 4: C_{4mm}(4) = 4.
- The set of symmetry operations of the point group 4mm which commute with m_{[100]} is {1, 2, m_{[100]} and m_{[010]}}. The centralizer of the m_{[100]} reflection with respect to the point group 4mm is the subgroup mm2 obtained by taking the two mirror reflections normal to the tetragonal a and b axes: C_{4mm}(m_{[100]}) = mm2.