Selected exam questions 2001-4
“Group theory: the best thing since sliced bread”
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Group Theory |
Sliced Loaf |
|---|---|
|
element of G |
crumb |
|
subgroup H of G |
chosen slice of bread |
|
coset Hg of H |
any slice of bread |
|
Lemma: |Hg| = |H| for all g in G |
Lemma: every slice has the same number of crumbs |
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Lemma: the cosets of H partition G |
Lemma: every crumb is in exactly one slice |
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Theorem: |G| = |H| x the number of cosets |
Theorem: the number of crumbs in the loaf = the number of crumbs in a slice x the number of slices |
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Space of cosets G/H |
Space of slices |
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If H is normal in G, quotient group G/H |
Group structure on space of slices |