This figure has four symmetry operations : the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes.
2.
The division operator reduces to the identity operation ( i . e ., field division is only defined for dividing by 1, and x / 1 = x ).
3.
C 1 is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter "'F " '.
4.
If we write " e " for " leave the blocks as they are " ( the identity operation ), then we can write the six permutations of the three blocks as follows:
5.
This expression of the identity operation is called a " representation " or a " resolution " of the identity ., This formal representation satisfies the basic property of the identity:
6.
D 1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter "'A " '.
7.
But this definition is not strong enough & mdash; if the possible messages are 0 and 1, D = { 0, 1 } and H consists of the identity operation and " not ", H is universal.
8.
A vector space consists of a set of elements together with an addition and scalar multiplication operation satisfying certain properties generalizing the properties of the familiar real vector spaces; for the cycle space, the elements of the vector space are the Eulerian subgraphs, the addition operation is symmetric differencing, multiplication by the scalar 1 is the identity operation, and multiplication by the scalar 0 takes every element to the empty graph, which forms the additive identity element for the cycle space.
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