Donald J Herzog Popular Books

Donald J Herzog Biography & Facts

In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.Let G{\displaystyle G} be a group, and let A={a1G1, …, akGk}{\displaystyle A=\{a_{1}G_{1},\ \ldots ,\ a_{k}G_{k}\}}be a finite system of left cosets of subgroups G1,…,Gk{\displaystyle G_{1},\ldots ,G_{k}} of G{\displaystyle G}. Herzog and Schönheim conjectured that if A{\displaystyle A} forms a partition of G{\displaystyle G} with k>1{\displaystyle k>1}, then the (finite) indices [G:G1],…,[G:Gk]{\displaystyle [G:G_{1}],\ldots ,[G:G_{k}]} cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if H{\displaystyle H} is any subgroup of G{\displaystyle G} with index k=[G:H]<∞{\displaystyle k=[G:H]<\infty } then G{\displaystyle G} can be partitioned into k{\displaystyle k} left cosets of H{\displaystyle H}. Subnormal subgroups In 2004, Zhi-Wei Sun proved an extended version of the Herzog–Schönheim conjecture in the case where G1,…,Gk{\displaystyle G_{1},\ldots ,G_{k}} are subnormal in G{\displaystyle G}. A basic lemma in Sun's proof states that if G1,…,Gk{\displaystyle G_{1},\ldots ,G_{k}} are subnormal and of finite index in G{\displaystyle G}, then [G:⋂i=1kGi] | ∏i=1k[G:Gi]{\displaystyle {\bigg [}G:\bigcap _{i=1}^{k}G_{i}{\bigg ]}\ {\bigg |}\ \prod _{i=1}^{k}[G:G_{i}]}and hence P([G:⋂i=1kGi] )=⋃i=1kP([G:Gi]),{\displaystyle P{\bigg (}{\bigg [}G:\bigcap _{i=1}^{k}G_{i}{\bigg ]}\ {\bigg )}=\bigcup _{i=1}^{k}P([G:G_{i}]),}where P(n){\displaystyle P(n)} denotes the set of prime divisors of n{\displaystyle n}. Mirsky–Newman theorem When G{\displaystyle G} is the additive group Z{\displaystyle \mathbb {Z} } of integers, the cosets of G{\displaystyle G} are the arithmetic progressions. In this case, the Herzog–Schönheim conjecture states that every covering system, a family of arithmetic progressions that together cover all the integers, must either cover some integers more than once or include at least one pair of progressions that have the same difference as each other. This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a regular polygon are colored in such a way that every color class itself forms the vertices of a regular polygon. Then, there exist two color classes that form congruent polygons. References. Discover the Donald J Herzog popular books. Find the top 100 most popular Donald J Herzog books.

Best Seller Donald J Herzog Books of 2024

  • Never Enough synopsis, comments

    Never Enough

    Lexy Timms

    Be good enough never is... Anthony Accardi is a man on a mission: make his father's watch company a success while bringing in millions of dollars. To do that, he needs an assi...

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    The Cupcake Cottage

    Jean Oram

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    Dracula

    Bram Stoker

    An Apple Books Classic edition. Few characters have seized readers’ imaginations quite like Count Dracula of Transylvania, the hero of Bram Stoker’s classic. The 1897 novel put vam...

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    Whiskey Girl

    Melissa Belle

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    The Odyssey

    Homer

    An Apple Books Classic edition. Homer’s eighthcentury epic poem is a companion to The Iliad . It tells the story of Odysseus, who journeys by ship for 10 years after the Trojan War...

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    Alpha

    Sybil Bartel

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    Nothing to Hide

    Scarlett Finn

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    Silenced Girls

    Roger Stelljes

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    The Wonderful Wizard of Oz

    L. Frank Baum

    An Apple Books Classic edition. You’ve seen the iconic 1939 movie, but do you know about the talking field mice, the Winkies, and the Witch of the North that appear in the original...

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    Teased by Fire

    Molly O'Hare

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    The Seduction Series Boxset

    Roxy Sloane

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    Escape, A New Life

    David J Antocci

    To save herself, she had to lose everything. Trapped in a tropical paradise with no memory of how she got there, Abby is thrust into a fight for her life. Hunted by a madman, and c...

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    A Bookshop to Die For

    M.P. Black

    Ditching her fiancé at the altar, Alice Hartford bolts to her childhood hometown to reconnect with the last, happy remnant of her past: her mom's old bookstore. But the bookst...

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    Think and Grow Rich

    Greg Habstritt & Napoleon Hill

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    Awaken Me

    Jenna Jacob

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    Eternal

    W.J. May

    She will fight for what is hers. When the king is murdered, Katerina, his only daughter, must flee for her life. She finds herself on a strange and dangerous path. Alone for the fi...

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    Pride and Prejudice

    Jane Austen

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    Man In The Water

    Jon Hill

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    His Girl Next Door

    Khardine Gray

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    Always Yours

    Claire Raye

    Some things are just meant to be... Ellen Somerville and Will McIntyre met by accident and under unusual circumstances. Getting sprayed by a skunk in a parking lot wouldn’t normall...

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    Finding Cinderella

    Colleen Hoover

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    You Are Kind

    Michael Gordon

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    All Fired Up

    Kathryn Shay

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    Dark Psychology and Manipulation

    Margaret Morrison

    THE MENTAL MANIPULATOR WILL NO LONGER KEEP SECRETS FROM YOU! Are you fed up with the wool being pulled over your eyes?Are you prepared to stand up to those who believe they can man...

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    Masters of Restraint

    Ines Johnson

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    Get Lucky

    Lila Monroe

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    Salvation

    Meghan O'Flynn

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    Wuthering Heights

    Emily Brontë

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    Coffee Girl

    Sophie Sinclair

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  • Holy Bible synopsis, comments

    Holy Bible

    The Church of Jesus Christ of Latter-day Saints

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    Assisting the Bosshole

    Kristin MacQueen

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    Meditations

    Emperor of Rome Marcus Aurelius

    Meditations is a series of personal writings by Marcus Aurelius, Roman Emperor 161–180 CE, setting forth his ideas on Stoic philosophy.

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    Becoming Lady Dalton

    Carrie Lomax

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    His Own Heaven

    Jennie Kew

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    The Honeymoon Homicide

    J. R. Mathis & Susan Mathis

    Enjoy this SmallTown Murder Mystery Featuring A Unique Sleuthing Couple I'm Father Tom Greer, a Catholic Priest in a smalltown parish who never expected this . . . When I came...

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    Inception of Gold

    Lexy Timms

    Things are not as bad as they seem. They are worse. Being the personal assistant to a narcissistic actress who just happens to own the key to my family's undoing isn't ex...

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    Silver Santa

    Lacey Silks

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    Good Guy

    Kate Meader

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    Rogue Alpha

    Kimber White

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    Pretty Little Lies

    Ivy Thorn

    Four years ago, Nicolo Marchetti took my innocence. But he left me with something else Nicolo almost took everything from me. My virginity, my reputation, and very nearly my future...

  • Christmas in Sweetbriar Cove synopsis, comments

    Christmas in Sweetbriar Cove

    Melody Grace

    Celebrate the holidays in Sweetbriar Cove with this festive romance collection, containing two sizzling smalltown holiday stories perfect for fans of Tessa Bailey, Sophie Kinsella...

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    Bewitching a Highlander

    Roma Cordon

    Defying all for the love of a bewitching lass. Breena MacRae, a healer from Skye with a touch of witchery in her blood, embarks on a dangerous search for her missing father. She ar...

  • The Count of Monte Cristo synopsis, comments

    The Count of Monte Cristo

    Alexandre Dumas

    An Apple Books Classic edition. Alexandre Dumas’ classic paints a portrait of Edmond Dantès, a dark and calculating man who is willing to wait years to exact his perfect plan for r...

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    Law of Attraction

    Jordan Hollis

    How to Finally Overcome the Hurdles of Manifesting Proven, effective and enjoyable ways to help you manifest faster… Right now, think of something that makes you successful. If you...

  • Enemies With Benefits synopsis, comments

    Enemies With Benefits

    Roxie Noir

    I don’t love him. I don’t even like him. I just want him. Eli Loveless was my nemesis from the first day of kindergarten until we graduated high school. Everything I did, he had to...

  • Little Women synopsis, comments

    Little Women

    Louisa May Alcott

    An Apple Books Classic edition. Meet the Marches! Louisa May Alcott’s classic introduces us to four unforgettable sisters: beautiful Meg, tomboyish Jo, delicate Beth, and Amy, the ...

  • The Stoic Mind synopsis, comments

    The Stoic Mind

    Addy Osmani & GoLimitlesss

    Discover the timeless wisdom of Stoicism in a modern context with "The Stoic Mind," an enlightening visual guide by Addy Osmani and GoLimitlesss. This rich exploration co...

  • The Icing on the Cake synopsis, comments

    The Icing on the Cake

    Linda Seed

    She’s a baker without a kitchen. He’s got a double oven to spare. It’s a recipe for success, until his mother starts stirring up trouble … Cassie Jordan has a sweet dream to open a...

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    Noxious

    Lexy Timms

    Stop setting yourself on fire to keep someone else warm. Brady and Levi have been together since high school, since before he became famous and started thinking only about himself....

  • Dream Psychology synopsis, comments

    Dream Psychology

    Sigmund Freud

    An Apple Books Classic edition. Written by the founding father of psychoanalysis, Sigmund Freud’s 1899 book is the definitive text on learning to interpret dreams. Freud’s groundbr...