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Group Theory Introduction

    The abstract definition of a group was not immediate to beginning group theorists. Indeed its formalization was slow to evolve since many mathematicians were interested in solving particular problems of the day. For example, Galois was interested in solving polynomial equations and even though he used group computations explicitly he did not actually define a group; consequently, his work was hard to read. Nevertheless, the notion of an abstract group began its journey with Cauchy, Cayley, Kronecker, Burnside, Weber and many others who led us to the definition we recognize today. Yet, one of the commonly given axioms for a group is debatable even today.
    An axiom sometimes called “closure” is really a consequence of the definition of a binary operation. Since we have already rigorously defined mappings and operations we will not include this axiom in the definition of a group. But it is important to realize that, by definition, groups must be, among other requirements, closed under the given operation.
    This topic gives the abstract definition of a group and then explains that groups and solving linear equations ( a x = b ) are the same. Some elementary properties and notation are then explained before giving many examples of some common groups.

A Definition of a Group

Definition (Group) A group is a set group theory introduction _gr_1.gif] together with an operation group theory introduction _gr_2.gif] on group theory introduction _gr_3.gif] such that

    (i) Associativity: group theory introduction _gr_4.gif] for all group theory introduction _gr_5.gif]
    
    (ii) Identity: there exists group theory introduction _gr_6.gif] such that group theory introduction _gr_7.gif] for all group theory introduction _gr_8.gif] and
    
    (iii) Inverses: for all group theory introduction _gr_9.gif] there exists group theory introduction _gr_10.gif] such that group theory introduction _gr_11.gif]
    
and is denoted by group theory introduction _gr_12.gif]

Definition (Abelian Group) An Abelian group is a group group theory introduction _gr_13.gif] such that (commutative) group theory introduction _gr_14.gif] for all group theory introduction _gr_15.gif]

Proposition (Unique Identity and Inverses) The identity element of a group group theory introduction _gr_16.gif] is unique and each element in a group has a unique inverse and is denoted by group theory introduction _gr_17.gif] for each group theory introduction _gr_18.gif]

    Proof. Assume that group theory introduction _gr_19.gif] and group theory introduction _gr_20.gif] are identity elements of group theory introduction _gr_21.gif] with respect to group theory introduction _gr_22.gif] Since group theory introduction _gr_23.gif] holds for all group theory introduction _gr_24.gif] replace group theory introduction _gr_25.gif] by group theory introduction _gr_26.gif] thus group theory introduction _gr_27.gif] Similarly,   group theory introduction _gr_28.gif] holds for all group theory introduction _gr_29.gif] replace group theory introduction _gr_30.gif] by group theory introduction _gr_31.gif] thus group theory introduction _gr_32.gif] Therefore, group theory introduction _gr_33.gif] and so every identity element must be the same, if there is one.
    Suppose group theory introduction _gr_34.gif] is a group with identity element group theory introduction _gr_35.gif] and group theory introduction _gr_36.gif] has inverses group theory introduction _gr_37.gif] and group theory introduction _gr_38.gif] then group theory introduction _gr_39.gif] group theory introduction _gr_40.gif] group theory introduction _gr_41.gif] group theory introduction _gr_42.gif]

Basic Properties of a Group

Proposition (Cancellation Property) Let group theory introduction _gr_43.gif] be a group and let group theory introduction _gr_44.gif]

    (i) Left Cancellation: If group theory introduction _gr_45.gif] then group theory introduction _gr_46.gif]
    
    (ii) Right Cancellation: If group theory introduction _gr_47.gif] then group theory introduction _gr_48.gif]
    
    Proof. (i): Suppose group theory introduction _gr_49.gif] Then group theory introduction _gr_50.gif] group theory introduction _gr_51.gif] yields group theory introduction _gr_52.gif] group theory introduction _gr_53.gif] Thus, group theory introduction _gr_54.gif] and so group theory introduction _gr_55.gif]
    (ii): Suppose group theory introduction _gr_56.gif] Then group theory introduction _gr_57.gif] group theory introduction _gr_58.gif] yields group theory introduction _gr_59.gif] group theory introduction _gr_60.gif] Thus, group theory introduction _gr_61.gif] and so group theory introduction _gr_62.gif] group theory introduction _gr_63.gif]

Proposition (Group Properties) Let group theory introduction _gr_64.gif] be a group with identity group theory introduction _gr_65.gif]

    (i) If group theory introduction _gr_66.gif] for some group theory introduction _gr_67.gif] then group theory introduction _gr_68.gif]
    
    (ii) If group theory introduction _gr_69.gif] then group theory introduction _gr_70.gif]
    
    (iii) If group theory introduction _gr_71.gif] then group theory introduction _gr_72.gif]
    
    (iv) group theory introduction _gr_73.gif] is Abelian if group theory introduction _gr_74.gif] for all group theory introduction _gr_75.gif]
    
    (v) group theory introduction _gr_76.gif] is Abelian if and only if group theory introduction _gr_77.gif] for all group theory introduction _gr_78.gif]
    
    Proof. (i): Since group theory introduction _gr_79.gif] the right cancellation property says that group theory introduction _gr_80.gif]
    (ii): Note group theory introduction _gr_81.gif] and group theory introduction _gr_82.gif] and thus group theory introduction _gr_83.gif] since inverses are unique (or by using left cancellation).
    (iii): Note that group theory introduction _gr_84.gif] and group theory introduction _gr_85.gif] and so   group theory introduction _gr_86.gif] since inverses are unique.
    (iv): If group theory introduction _gr_87.gif] for all group theory introduction _gr_88.gif] and since group theory introduction _gr_89.gif] for all group theory introduction _gr_90.gif] it follows that group theory introduction _gr_91.gif] for all group theory introduction _gr_92.gif] because inverses are unique. So, group theory introduction _gr_93.gif] group theory introduction _gr_94.gif] group theory introduction _gr_95.gif] which proves that group theory introduction _gr_96.gif] is Abelian.
    (v): If group theory introduction _gr_97.gif] is Abelian, then group theory introduction _gr_98.gif] Conversely, assume   group theory introduction _gr_99.gif] for all group theory introduction _gr_100.gif] Then group theory introduction _gr_101.gif] group theory introduction _gr_102.gif] group theory introduction _gr_103.gif] group theory introduction _gr_104.gif] group theory introduction _gr_105.gif]

    Let group theory introduction _gr_106.gif] be a group. Sometimes a group operation group theory introduction _gr_107.gif] is denoted using standard multiplication notation; for example, group theory introduction _gr_108.gif] might be denoted by group theory introduction _gr_109.gif] or even group theory introduction _gr_110.gif] The unique inverse of group theory introduction _gr_111.gif] is denoted by group theory introduction _gr_112.gif] The product group theory introduction _gr_113.gif] is denoted by group theory introduction _gr_114.gif] The   group theory introduction _gr_115.gif] factors in the product group theory introduction _gr_116.gif] is denoted by group theory introduction _gr_117.gif] The identity of group theory introduction _gr_118.gif] is sometimes denoted by group theory introduction _gr_119.gif] and also by group theory introduction _gr_120.gif] The notation group theory introduction _gr_121.gif] means group theory introduction _gr_122.gif] or group theory introduction _gr_123.gif]
    Sometimes a group operation group theory introduction _gr_124.gif] is denoted using standard addition notation (especially if the group is Abelian); for example, group theory introduction _gr_125.gif] might be denoted by group theory introduction _gr_126.gif] The unique inverse of group theory introduction _gr_127.gif] is denoted by group theory introduction _gr_128.gif] The sum group theory introduction _gr_129.gif] is denoted by group theory introduction _gr_130.gif] The group theory introduction _gr_131.gif] terms in the sum group theory introduction _gr_132.gif] is denoted by group theory introduction _gr_133.gif] The identity of group theory introduction _gr_134.gif] is sometimes denoted by group theory introduction _gr_135.gif] and also by group theory introduction _gr_136.gif] The notation group theory introduction _gr_137.gif] means group theory introduction _gr_138.gif] or group theory introduction _gr_139.gif]

Proposition (Induction Properties) Let group theory introduction _gr_140.gif] be a group with group theory introduction _gr_141.gif]. Then

    (i) group theory introduction _gr_142.gif] for all group theory introduction _gr_143.gif]
    
    (ii) group theory introduction _gr_144.gif] for all group theory introduction _gr_145.gif] and
    
    (iii) group theory introduction _gr_146.gif] for all group theory introduction _gr_147.gif]
    
    Proof. (i): Let group theory introduction _gr_148.gif] and prove that group theory introduction _gr_149.gif] for all group theory introduction _gr_150.gif] and all positive integers group theory introduction _gr_151.gif] by induction. If group theory introduction _gr_152.gif] then group theory introduction _gr_153.gif] If group theory introduction _gr_154.gif] then group theory introduction _gr_155.gif] group theory introduction _gr_156.gif] group theory introduction _gr_157.gif] group theory introduction _gr_158.gif] group theory introduction _gr_159.gif] group theory introduction _gr_160.gif] For group theory introduction _gr_161.gif] group theory introduction _gr_162.gif] group theory introduction _gr_163.gif] group theory introduction _gr_164.gif] group theory introduction _gr_165.gif] If group theory introduction _gr_166.gif] then group theory introduction _gr_167.gif] group theory introduction _gr_168.gif] group theory introduction _gr_169.gif] group theory introduction _gr_170.gif] group theory introduction _gr_171.gif] group theory introduction _gr_172.gif] as required.    
    Next prove that group theory introduction _gr_173.gif] for any group theory introduction _gr_174.gif] and for group theory introduction _gr_175.gif] If group theory introduction _gr_176.gif] for all group theory introduction _gr_177.gif] then group theory introduction _gr_178.gif] group theory introduction _gr_179.gif] group theory introduction _gr_180.gif] group theory introduction _gr_181.gif] group theory introduction _gr_182.gif] group theory introduction _gr_183.gif] Now suppose group theory introduction _gr_184.gif] for any group theory introduction _gr_185.gif] Then group theory introduction _gr_186.gif] group theory introduction _gr_187.gif] group theory introduction _gr_188.gif] group theory introduction _gr_189.gif] Thus, group theory introduction _gr_190.gif] for all group theory introduction _gr_191.gif]  as required.    
    (ii): First, proof by induction on group theory introduction _gr_192.gif] for group theory introduction _gr_193.gif] Asume group theory introduction _gr_194.gif] for any group theory introduction _gr_195.gif] then group theory introduction _gr_196.gif] group theory introduction _gr_197.gif] group theory introduction _gr_198.gif] group theory introduction _gr_199.gif] group theory introduction _gr_200.gif] and so group theory introduction _gr_201.gif] holds for group theory introduction _gr_202.gif] and group theory introduction _gr_203.gif] Next, if group theory introduction _gr_204.gif] and group theory introduction _gr_205.gif] then group theory introduction _gr_206.gif] group theory introduction _gr_207.gif] group theory introduction _gr_208.gif] group theory introduction _gr_209.gif]  as required.    
    (iii): For any group theory introduction _gr_210.gif] group theory introduction _gr_211.gif]  as required. group theory introduction _gr_212.gif]

A Characterization of a Group

Proposition (Group Characterization) If group theory introduction _gr_213.gif] is a group and group theory introduction _gr_214.gif] then the equations group theory introduction _gr_215.gif] and group theory introduction _gr_216.gif] have unique solutions. Conversely, if group theory introduction _gr_217.gif] is a nonempty set with an associative binary operation in which the equations group theory introduction _gr_218.gif] and group theory introduction _gr_219.gif] have solutions for any group theory introduction _gr_220.gif] then group theory introduction _gr_221.gif] is a group.

    Proof.     The equations group theory introduction _gr_222.gif] and group theory introduction _gr_223.gif] have solutions, namely group theory introduction _gr_224.gif] and group theory introduction _gr_225.gif] respectively. Let group theory introduction _gr_226.gif] be another solution for the equation group theory introduction _gr_227.gif] Then multiplying both sides of the equation by group theory introduction _gr_228.gif] yields group theory introduction _gr_229.gif] and so group theory introduction _gr_230.gif] Since inverses are unique group theory introduction _gr_231.gif] is the only solution. Similarily, group theory introduction _gr_232.gif] is the only solution to group theory introduction _gr_233.gif]
     Conversely, assume the equations group theory introduction _gr_234.gif] and group theory introduction _gr_235.gif] have  solutions for all group theory introduction _gr_236.gif] where group theory introduction _gr_237.gif] is an associative operation on group theory introduction _gr_238.gif] In particular, group theory introduction _gr_239.gif] has a solution, say group theory introduction _gr_240.gif] for some group theory introduction _gr_241.gif] In fact group theory introduction _gr_242.gif] is the identity of group theory introduction _gr_243.gif] To see this let group theory introduction _gr_244.gif] and let group theory introduction _gr_245.gif] be any solution to the equation group theory introduction _gr_246.gif] then group theory introduction _gr_247.gif] group theory introduction _gr_248.gif] group theory introduction _gr_249.gif] group theory introduction _gr_250.gif] Let group theory introduction _gr_251.gif] group theory introduction _gr_252.gif] is a solution to group theory introduction _gr_253.gif] and group theory introduction _gr_254.gif] be any solution to the equation group theory introduction _gr_255.gif] then group theory introduction _gr_256.gif] group theory introduction _gr_257.gif] group theory introduction _gr_258.gif] group theory introduction _gr_259.gif] But then group theory introduction _gr_260.gif] and group theory introduction _gr_261.gif] and so group theory introduction _gr_262.gif] Thus group theory introduction _gr_263.gif] satisfies the requirements of being the identity element on group theory introduction _gr_264.gif] Finally, given any group theory introduction _gr_265.gif] let group theory introduction _gr_266.gif] be a solution of the equation group theory introduction _gr_267.gif] and let group theory introduction _gr_268.gif] be a solution of the equation group theory introduction _gr_269.gif] Then group theory introduction _gr_270.gif] group theory introduction _gr_271.gif] group theory introduction _gr_272.gif] group theory introduction _gr_273.gif] group theory introduction _gr_274.gif] and so group theory introduction _gr_275.gif] Thus group theory introduction _gr_276.gif] and group theory introduction _gr_277.gif] and so group theory introduction _gr_278.gif] is an inverse for group theory introduction _gr_279.gif] Therefore, group theory introduction _gr_280.gif] is a group as required. group theory introduction _gr_281.gif]

Definition (Finite Groups) A finite group is a group with a finite number of elements and the number of elements is the order of the group.

Proposition (Group Operation Table) Let group theory introduction _gr_282.gif] be a group. Then each element occurs exactly once as an entry on each row and each column of the table for the operation of a finite group group theory introduction _gr_283.gif]

    Proof. The entries on the row for an element group theory introduction _gr_284.gif] are group theory introduction _gr_285.gif] group theory introduction _gr_286.gif]..., group theory introduction _gr_287.gif] Any element group theory introduction _gr_288.gif] of group theory introduction _gr_289.gif] appears as the entry group theory introduction _gr_290.gif] with group theory introduction _gr_291.gif] No element can appear more than once since that would mean that group theory introduction _gr_292.gif] with group theory introduction _gr_293.gif] but this is impossible because of left cancellation. The same applies for columns by the cancellation law. group theory introduction _gr_294.gif]

Some Examples of Groups

Example (Common Groups)

    (i) The integers, rational numbers, real numbers, and complex numbers, are all Abelian groups with respect to ordinary addition.
    
    (ii) The nonzero rational numbers is an Abelian group with respect to ordinary multiplication.
    
    (iii) The set of even (odd) integers with ordinary addition is an Abelian group.
    
    (iv) Let group theory introduction _gr_295.gif] be a set, then the set of all invertible mappings group theory introduction _gr_296.gif] from group theory introduction _gr_297.gif] to group theory introduction _gr_298.gif] is a group with composition as the operation.
    
    (v) The set group theory introduction _gr_299.gif] of all group theory introduction _gr_300.gif] real matrices together with matrix addition as the operation is an Abelian group.
    
    (vi) The set group theory introduction _gr_301.gif] of all group theory introduction _gr_302.gif] real matices with nonzero determinant together with matrix multiplication as the operation is a non-Abelian group and is named the general linear group of   group theory introduction _gr_303.gif] matrices over group theory introduction _gr_304.gif]
    
    (vii) The set group theory introduction _gr_305.gif] of all real-valued continuous functions on group theory introduction _gr_306.gif] with pointwise addition (function addition) defined by group theory introduction _gr_307.gif] is an Abelian group.  More generally, the set of all mappings from group theory introduction _gr_308.gif] to group theory introduction _gr_309.gif] with function addition is an Abelian group; and also the set
    
             group theory introduction _gr_310.gif]    
                
is a group under function multiplication.

    (viii) The set group theory introduction _gr_311.gif] consisiting of all subsets of a given set group theory introduction _gr_312.gif] with operation defined by group theory introduction _gr_313.gif] is an Abelian group.
    
    (ix) The set group theory introduction _gr_314.gif] with complex multiplication is a group. In general, for group theory introduction _gr_315.gif] the set group theory introduction _gr_316.gif] of complex roots of unity
    
             group theory introduction _gr_317.gif]
                
is an Abelian group under complex multiplication.

    (x) The set group theory introduction _gr_318.gif] with componentwise addition is a group. In general, vector spaces are groups under vector addition.
    
    (xi) The set
    
group theory introduction _gr_319.gif]

is an Abelian group under matrix multiplcation.

    (xii) The following matrices form a group under matrix multiplication and is called the quaternion group:
    
             group theory introduction _gr_320.gif] group theory introduction _gr_321.gif] group theory introduction _gr_322.gif] group theory introduction _gr_323.gif]
        
              group theory introduction _gr_324.gif] group theory introduction _gr_325.gif] group theory introduction _gr_326.gif]     group theory introduction _gr_327.gif]
group theory introduction _gr_328.gif]

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Group Theory Introduction
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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