Abstract Huge
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Abstract Huge
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 JOAN MIROFAB COLOR LITHO LIMITED EDITION of 300 $156.49 (1 Bid) Time Remaining: 3h |
 MODERN ABSTRACT HUGE WALL ART OIL PAINTING ON CANVAS $0.01 Time Remaining: 1h 2m |
 Donald Sultan Original Hand signed Screenprint $405.00 (11 Bids) Time Remaining: 6h 57m |
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 Donald Sultan Original Hand signed Screenprint $500.00 (14 Bids) Time Remaining: 6h 59m |
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 ALEXANDER CALDER FAB LARGE ORIGINAL LITHOGRAPH 1966 NO RESERV $39.52 Time Remaining: 5h 23m |
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 ANDY WARHOL Ltd EDITION LITHOGRAPH SOUP $94.84 (2 Bids) Time Remaining: 4h 50m |
 Asian style Modern Abstract Huge Art Oil Painting Canvas Large Tree+free gift $42.88 Time Remaining: 11d 12h 43m Buy It Now for only: $42.88 Buy It Now |
 Pencil Signed Mystery Artist Composition On Paper From Venezuelan ESTATE Lot 45 $0.01 (1 Bid) Time Remaining: 11h 14m |
 1 of 2 huge ABSTRACT oil indistincly signed VELDE NO RES $16.50 (2 Bids) Time Remaining: 9h 59m |
 Signed Rene Portocarrero 1912 1986 Gouache Oil On Paper Painting CUBA 106 $460.00 (16 Bids) Time Remaining: 12h 54m |
 Modern Abstract Huge Wall Art Oil Painting On Canvas 08 $45.00 Time Remaining: 3d 16h 2m Buy It Now for only: $45.00 Buy It Now |
 HUGE Bronze Abstract Emaus Sun Sol Sculpture Mid Century Modern Eames 1960s $399.00 (1 Bid) Time Remaining: 16h 1m |
 Pencil Signed 1969 Karel Appel 1921 2006 Personnage LE Color Lithograph 226 $305.00 (7 Bids) Time Remaining: 15h 9m |
 2 of 2 huge ABSTRACT oil indistincly signed VELDE NO RESERVE $9.95 (1 Bid) Time Remaining: 10h 4m |
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 LISTED Signed Jonas Gerard 1977 Madona Of The Sea Oil On Board Painting NR 30 $5.50 (2 Bids) Time Remaining: 11h |
 Artist Signed Bill Dodd Metal Abstract Acrobat Formalist Sculpture Base Incised $31.00 (8 Bids) Time Remaining: 16h 9m |
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 Abstract Huge Art Oil Painting wall decoratno framed 051301 $0.96 Time Remaining: 1h 12m |
 Vintage Pencil Signed Original PETER MAX Mid Century 93 300 Lithograph Lot 32 $200.00 (8 Bids) Time Remaining: 11h 5m |
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 SIGNED Jose Ramon Oil on Canvas Abstract Expressionist Painting Cuba Lot 68 $11.50 (4 Bids) Time Remaining: 12h 9m |
 Pencil Signed Mystery Artist Abstract Composition Embossed Etching ESTATE 142 $21.50 (4 Bids) Time Remaining: 14h 2m |
 Signed Wifredo Oscar de la Concepcion Lam 1902 1982 Mixed Media On Paper 280 $102.50 (9 Bids) Time Remaining: 16h 21m |
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 Mystery Signed Abstract Assembled Glass Geometric Sculpture Mid Century Lot 277 $78.77 (10 Bids) Time Remaining: 16h 3m |
 Signed Dan Murphy Abstract Metal Sculpture w Lucite Base Mid Century LISTED 276 $81.00 (14 Bids) Time Remaining: 16h 22m |
 MODERN ABSTRACT HUGE LARGE CANVAS ART OIL PAINTING 051308 $0.83 Time Remaining: 1h 20m |
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 ELLSWORTH KELLY FAB LARGE ORIGINAL LITHOGRAPH 1964 NO RESERV $31.61 Time Remaining: 5h 5m |
 LISTED Signed Stefan Alexander Figural Abstract Mid Century Oil Painting 119 $56.99 (11 Bids) Time Remaining: 13h 27m |
 Heinz Aeschlimann Contemporary Swiss Metal Kinetic Geometric Sculpture Lot 302 $306.00 (19 Bids) Time Remaining: 14h 37m |
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 Signed George Aarons 1896 1980 Figural Plaster Relief LISTED England Lot 260 $13.51 (6 Bids) Time Remaining: 15h 57m |
 LISTED Purvis Young 1943 2010 Outsider Art JAIL Painting On Wood Unsigned 185 $67.00 (17 Bids) Time Remaining: 13h 45m |
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 SIGNED Mystery Artist Abstract Oil On Masonite Painting Mid Century 327 $36.00 (7 Bids) Time Remaining: 14h 52m |
 Modern art Abstract Huge Oil Painting No Frame $0.50 Time Remaining: 1h 41m |
 SIGNED Peter Robert Keil 1942 Oil On Canvas Painting Birth Of Judas 62 217 $201.49 (13 Bids) Time Remaining: 14h 12m |
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 60 X 90 cm Huge Canvas Wall Art Abstract Oil Painting Modern Decor Hand Painted $29.49 (1 Bid) Time Remaining: 30m |
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 LISTED Signed Alexander Gore 1958 USA Russian Fed Oil Painting FRAMED 199 $86.00 (17 Bids) Time Remaining: 14h 52m |
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 SIGNED Red Grooms 1937 Three Dimensional Mix Media Paper Mach Oil Painting 121 $2,070.00 (13 Bids) Time Remaining: 14h 1m |
 HUGE MODERN ABSTRACT PAINTING FINE ART60X24X15ELOISExxx $49.00 Time Remaining: 16h 38m |
 Pencil Signed Victor Vasarely 1908 1997 Composition Lithograph Litho Lot 61 $162.50 (26 Bids) Time Remaining: 12h |
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 RETRO MID CENTURY ABSTRACT ART DRIP PAINTING INFLUENCED BY JACKSON POLLOCK $39.00 Time Remaining: 2h 8m |
 Important 62 Peter Keil 1942 Abstract Composition Oil On Canvas Painting 218 $204.50 (9 Bids) Time Remaining: 14h 31m |
 Signed Purvis Young 1943 2010 Outsider Art TRUCK Painting On Wood LISTED 180 $112.50 (19 Bids) Time Remaining: 10h 46m |
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 70s Mid Century Henry Moretti Abstract Charcoal Pastel Etching SIGNED HM Lot 69 $10.50 (4 Bids) Time Remaining: 12h 12m |
 ART MODERN ABSTRACT HUGE WALL DECORATE CANVAS ART OIL PAINTING $0.99 Time Remaining: 2h 7m |
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 Huge Original 60x 27 Acrylic Abstract Seascape Wall Art Painting On Canvas $135.00 Time Remaining: 1h 1m |
 LISTED Signed Alexander Gore 1958 USA Russian Fed Oil Painting FRAMED 201 $103.52 (19 Bids) Time Remaining: 16h 8m |
 Modern Abstract Huge Canvas Oil Painting $0.10 Time Remaining: 1h 35m |
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 Danish Mid Century Modern Art Fish Wall Sculpture Painting Retro Eames Era Tiki $227.50 (2 Bids) Time Remaining: 14h 23m |
 LISTED Signed Alexander Gore 1958 USA Russian Fed Oil Painting FRAMED 196 $79.00 (13 Bids) Time Remaining: 12h 45m |
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 LISTED Artist Signed Stefan Alexander Abstract Drawling Mid Century Print 100 $14.50 (3 Bids) Time Remaining: 12h 48m |
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 HANDMADE Modern Abstract Huge Oil Painting Canvas Large Art $0.10 Time Remaining: 1h 38m |
 Modern Abstract Huge Wall Art Oil Painting On Canvas $0.10 (1 Bid) Time Remaining: 1h 40m |
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 HANDMADE Modern Abstract Huge Oil Painting Canvas Large Art 3p $0.10 Time Remaining: 1h 39m |
 Roy Lichtenstein original pencil graphite drawing signed w COA Andy Warhol era $355.00 (5 Bids) Time Remaining: 9h 28m |
 Modern Abstract Huge Wall Art Oil Painting On Canvas No frame $1.99 Time Remaining: 29d 13h 41m Buy It Now for only: $1.99 Buy It Now |
 Important Peter Robert Keil 1942 Painting Pablo Picasso 1962 w COA 220 $406.00 (26 Bids) Time Remaining: 15h 11m |
 Modern Abstract Huge Wall Art Oil Painting On Canvas 4P $0.10 Time Remaining: 1h 40m |
 LARGE ORIGINAL FINE ART ABSTRACT MODERN OIL PAINTING WALL DECOR Eugenia Abramson $169.00 (1 Bid) Time Remaining: 14h 29m |
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 SIGNED Stig L Lindberg Swedish Abstract Screen Mid Century Wall Decor Lot 270 $36.00 (7 Bids) Time Remaining: 16h 20m |
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Abstract Algebra: Groups and properties of their elements... PLEASE HELP?
Here's two problems I need HUGE help with:
1. If G = { g1, g2, ..., gn} is an Abelian group, show that the product p = g1g2...gn of all elements of G equals the product of the elements in G of order 2.
2.Let a and b be elements of a group. Assume that |a| = m and |b| = n, where m and n are relatively prime. If ab=ba, show |ab| = mn.
I have absolutely no idea how to show question 1, and I didn't get too far with question 2. All I knew is that since m and n are relatively prime, I should use the fact that 1 = xm + yn for some integers x and y. I couldn't figure anything out even after playing with the facts that a^m = 1 and b^n = 1.
Please help with full proofs? This material is really tough for me 
1. this is easier than it looks.
suppose gi is an element NOT of order 1 or 2. then gi^-1 is some "other" element of G, and when the full product of every member of G is taken, these two elements will cancel out (we can re-arrange the elements of G in the product, to put the inverses next to each other, since G is abelian). so the only elements that won't cancel out in pairs, are the elements of order 2, and e (which makes no difference to the product).
2.first of all, if ab = ba, then (ab)^2 = abab = a(ba)b = a(ab)b = a^2b^2. in a similar vein (a rigorous proof needs induction), (ab)^n = (a^n)(b^n).
so (ab)^mn = (a^mn)(b^mn) = ((a^m)^n)((b^n)^m)) = (e^n)(e^m) = e
this at least shows |ab| divides mn.
now since (ab)^k = (a^k)(b^k), if (ab)^k = e, then both a^k = e, and b^k = e
(remember that |a| and |b| are relatively prime, so we can't have a^k = x, b^k = x^-1,
unless x = e, because if a^k = b^j, then |a^k| divides |a| and also divides |b| (since
a^k is also in ) meaning |a^k| is a common divisor of m and n, that is to say |a^k| = 1.)
thus |a| divides |ab| and |b| divides |ab|, so m and n divide |ab|, hence lcm(m,n) divides |ab|.
but m and n are relatively prime, so lcm(m,n) = mn, which divides |ab|.
(perhaps it will help to consider a group like Z12. in Z12, 6 has order 2, while 4 has order 3.
so what is the order of 10 ( = 6 + 4)?
(6 + 4) = 10
2(6 + 4) = 2(6) + 2(4) = 0 + 8 = 8
3(6 + 4) = 3(6) + 3(4) = 6 + 0 = 6
4(6 + 4) = 4(6) + 4(4) = 0 + 4 = 4
5(6 + 4) = 5(6) + 5(4) = 6 + 8 = 2
6(6 + 4) = 6(6) + 6(4) = 0 + 0 = 0, 10 has order 6 = 2*3.).