198:
is a CA-group if and only if it is abelian. The finite CA-groups are classified: the solvable ones are semidirect products of abelian groups by cyclic groups such that every non-trivial element acts fixed-point-freely and include groups such as the
178:
of an abelian group and a fixed-point-free automorphism, and that conversely every such semidirect product is a finite, solvable CA-group. Wu also extended the classification of Suzuki et al. to
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which states that every finite, simple, non-abelian group is of even order. A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in (
345:
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on 4 points of order 12, while the nonsolvable ones are all simple and are the 2-dimensional projective special linear groups PSL(2, 2) for
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91:
Locally finite CA-groups were classified by several mathematicians from 1925 to 1998. First, finite CA-groups were shown to be
116:
47:. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the
512:
391:
340:
262:
143:
265:; Wall, G. E. (1958), "A characterization of the one-dimensional unimodular projective groups over finite fields",
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48:
158:. This result was first extended to the Feit–Hall–Thompson theorem showing that finite, simple, non-abelian,
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179:
155:
68:
170:, pp. 291–305). A more detailed description of the Frobenius groups appearing is included in (
80:
25:
424:
Weisner, L. (1925), "Groups in which the normaliser of every element except identity is abelian",
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403:
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amongst the non-identity elements of a group if and only if the group is a CA-group.
41:
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398:, Grundlehren der Mathematischen Wissenschaften , vol. 248, Berlin, New York:
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21:
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235:, p. 10). Some more recent results in the infinite case are included in (
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Wu, Yu-Fen (1998), "Groups in which commutativity is a transitive relation",
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343:(1957), "The nonexistence of a certain type of simple groups of odd order",
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CA-groups have been classified explicitly. CA-groups are also called
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127:≥ 2. Finally, finite CA-groups of odd order were shown to be
55:. Several important infinite groups are CA-groups, such as
174:), where it is shown that a finite, solvable CA-group is a
135:), and so in particular, are never non-abelian simple.
247:are obvious examples of infinite simple CA-groups.
111:), finite CA-groups of even order were shown to be
346:Proceedings of the American Mathematical Society
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138:CA-groups were important in the context of the
194:is a CA-group, and a group with a non-trivial
427:Bulletin of the American Mathematical Society
8:
232:
477:
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358:
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100:
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140:classification of finite simple groups
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79:for short) because commutativity is a
53:classification of finite simple groups
115:, abelian groups, or two dimensional
7:
154:, non-abelian, CA-group is of even
243:CA-groups. Wu also observes that
236:
171:
14:
239:), including a classification of
40:of any nonidentity element is an
215:≥ 2. Infinite CA-groups include
162:had even order, and then to the
117:projective special linear groups
441:10.1090/S0002-9904-1925-04079-3
267:Illinois Journal of Mathematics
109:Brauer, Suzuki & Wall 1958
1:
123:of even order, PSL(2, 2) for
73:commutative-transitive groups
529:
312:Combinatorial group theory
231:of large prime exponent, (
105:Brauer–Suzuki–Wall theorem
34:centralizer abelian group
233:Lyndon & Schupp 2001
479:10.1006/jabr.1998.7468
280:10.1215/ijm/1255448336
131:or abelian groups in (
180:locally finite groups
164:Feit–Thompson theorem
49:Feit–Thompson theorem
513:Properties of groups
314:, Berlin, New York:
81:transitive relation
466:Journal of Algebra
176:semidirect product
146:showed that every
20:, in the realm of
409:978-0-387-10916-9
325:978-3-540-41158-1
209:alternating group
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396:Group theory. II
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304:Lyndon, Roger C.
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129:Frobenius groups
113:Frobenius groups
103:). Then in the
28:is said to be a
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400:Springer-Verlag
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360:10.2307/2033280
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316:Springer-Verlag
308:Schupp, Paul E.
302:
273:(4B): 718–745,
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245:Tarski monsters
229:Burnside groups
201:dihedral groups
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89:
65:Burnside groups
61:Tarski monsters
12:
11:
5:
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472:(1): 165–181,
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434:(8): 413–416,
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392:Suzuki, Michio
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353:(4): 686–695,
341:Suzuki, Michio
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263:Suzuki, Michio
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241:locally finite
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121:finite field
101:Weisner 1925
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22:group theory
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251:Works cited
217:free groups
168:Suzuki 1986
133:Suzuki 1957
63:, and some
57:free groups
38:centralizer
18:mathematics
458:51.0112.06
259:Brauer, R.
203:of order 4
67:, and the
488:0021-8693
450:0002-9904
369:0002-9939
289:0019-2082
160:CN-groups
77:CT-groups
507:Category
394:(1986),
310:(2001),
186:Examples
97:solvable
51:and the
45:subgroup
30:CA-group
496:1643082
418:0815926
385:0086818
377:2033280
334:0577064
297:0104734
237:Wu 1998
221:PSL(2,
172:Wu 1998
119:over a
87:History
42:abelian
36:if the
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227:, and
196:center
190:Every
152:simple
148:finite
93:simple
373:JSTOR
156:order
26:group
484:ISSN
446:ISSN
404:ISBN
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