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Definition
1) closure under some binary operation
2) associativity
3) contains the identity element
4) set contains the inverse of each element |
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| H is a group under the operation of G, all elements in H are contained in G |
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| example of a group that is not cyclic |
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| cyclic subgroup if G generated by a |
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Definition
| (a in brackets) ={a^n | n in Z <\bf>} |
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Definition
The center of a group G:
{a in G | ax=xa for all x in G}. |
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centralizer:
{g in G | ga=ag} |
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| the order of an element: the smallest possible n such that a^n=e. |
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Definition
Let G be a group and H be a non-empty subgroup of G. If a b^{-1} is in H whenever a and b are in h, then H is a subgroup of G.
(In additive notation, if a-b is in H whenever a and b are in H, then H is a subgroup of G.) |
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| Prove cancellation in a group. |
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Definition
(let a' be the inverse of a.)
Suppose ba=ca. Then (ba)a'=(ba)a' implies b(aa')=c(aa')implies b=c. |
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| Prove uniqueness of inverse in a group. |
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Definition
| Suppose b, c are both inverses of a. Then ab=e and ac=e, so that ab-ac. Cancelling a on both sides gives b=c. |
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write the following permutation as a product of transpositions:
(12345) |
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Definition
| (12345)=(12)(23)(34)(45)=(25)(15)(32)(43) |
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| always even or always odd theorem for permutations |
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Definition
| If a permutation alpha can be expressed as a product of an even/odd number of 2-cycles, then every decomposition of alpha into a product of 2-cycles must have an even/odd number of 2-cycles. |
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| fundamental theorem of cyclic groups |
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Definition
| Every subgroup of a cyclic group is cyclic. Moreover, if |a in brackets| =n, then the order of any subgroup of (a in brackets) is a divisor of n; and, for each positive divisor k of n, the group(a in brackets) has exactly one subgroup of order k - namely, ((a^{n/k}) in brackets) |
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multiply the following pairs of permutations in cycle notation:
(13)(284)(5679)(154978632)
(154978632)(1928)(3)(47)(56) |
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Definition
(13)(284)(5679)(154978632)=(16)(238745)
(154978632)(1928)(3)(47)(56)= (179)(264853) |
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| show that Z(G) is a subgroup of G |
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Definition
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| Show that C(a) is a subgroup of G |
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