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9 Cards in this Set
- Front
- Back
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Lagrange's Theorem |
For any finite group G, the order or any subgroup H divides the order of G/ |
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Cayley's Theorem |
Every group G is isomorphic to a subgroup of sym(G) |
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First Isomorphism Theorem |
Given a homo f:G->H a) The kernel of f is a normal subgroup of G b) The image of f is a subgroup of H c) im(f) is isomorphic to G/ker(f) |
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Second Isomorphism Theorm |
We don't care! |
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Third Isomorphism Theorem |
If N and K are normal subgroups of G such that K<N<G 1. N/K <|G/K 2. (G/K)/(N/K)=G/N |
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Group Action |
f:GxA=>A 1. f_g(f_h(a))=f_gh(a) 2. f_e(a)=a |
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Qutotient group |
G/N: the group of left cosets, is a group iff N is normal |
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Equivalencies of Normal Subgroups |
1. Normal subgroup 2. gN=Ng 3. The cosets coincide 4. gNg^-1 <= N (If finite, if infinite, = to N |
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Cauchy's Theorem |
If G is a finite group and p is prime dividing |G| then G has an element of order p. |
