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One of the two groups of Order 4. Like
, it is Abelian,
but unlike
, it is a Cyclic. Examples include the Point Groups
and
and the Modulo Multiplication Groups
and
. Elements
of
the group satisfy
, where 1 is the Identity Element, and two of the elements satisfy
.
The Cycle Graph is shown above. The Multiplication Table for this group may be written in three equivalent
ways--denoted here by ,
, and
--by permuting the symbols used for the group elements.
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1 | ![]() |
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1 | 1 | ![]() |
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1 |
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1 | ![]() |
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1 | ![]() |
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The Multiplication Table for is obtained from
by interchanging
and
.
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1 | ![]() |
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1 | 1 | ![]() |
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1 | ![]() |
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1 |
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1 | ![]() |
The Multiplication Table for is obtained from
by interchanging
and
.
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1 | ![]() |
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1 | 1 | ![]() |
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1 | ![]() |
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1 | ![]() |
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1 |
The Conjugacy Classes of are
,
,
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(1) |
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(2) |
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(3) |
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(4) |
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(5) |
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(6) |
The group may be given a reducible representation using Complex Numbers
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(7) |
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(8) |
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(9) |
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(10) |
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(11) |
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(12) |
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(13) |
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(14) |
See also Finite Group Z2Z2
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© 1996-9 Eric W. Weisstein