Introducing Groups

    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.

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

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

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

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

    Proof. That the identity element of a group is unique follows from the Unique Identity Element Proposition. Suppose introducing groups _gr_19.gif] is a group with identity element introducing groups _gr_20.gif] and introducing groups _gr_21.gif] has inverses introducing groups _gr_22.gif] and introducing groups _gr_23.gif] then introducing groups _gr_24.gif] introducing groups _gr_25.gif] introducing groups _gr_26.gif] introducing groups _gr_27.gif]

Proposition (Cancellation Property) Let introducing groups _gr_28.gif] be a group and let introducing groups _gr_29.gif]

    (i) Left Cancellation: If introducing groups _gr_30.gif] then introducing groups _gr_31.gif]
    
    (ii) Right Cancellation: If introducing groups _gr_32.gif] then introducing groups _gr_33.gif]
    
    Proof. (i): Suppose introducing groups _gr_34.gif] Then introducing groups _gr_35.gif] introducing groups _gr_36.gif] yields introducing groups _gr_37.gif] introducing groups _gr_38.gif] Thus, introducing groups _gr_39.gif] and so introducing groups _gr_40.gif]
    (ii): Suppose introducing groups _gr_41.gif] Then introducing groups _gr_42.gif] introducing groups _gr_43.gif] yields introducing groups _gr_44.gif] introducing groups _gr_45.gif] Thus, introducing groups _gr_46.gif] and so introducing groups _gr_47.gif] introducing groups _gr_48.gif]

Proposition (Group Properties) Let introducing groups _gr_49.gif] be a group with identity introducing groups _gr_50.gif]

    (i) If introducing groups _gr_51.gif] for some introducing groups _gr_52.gif] then introducing groups _gr_53.gif]
    
    (ii) If introducing groups _gr_54.gif] then introducing groups _gr_55.gif]
    
    (iii) If introducing groups _gr_56.gif] then introducing groups _gr_57.gif]
    
    (iv) introducing groups _gr_58.gif] is Abelian if introducing groups _gr_59.gif] for all introducing groups _gr_60.gif]
    
    (v) introducing groups _gr_61.gif] is Abelian if and only if introducing groups _gr_62.gif] for all introducing groups _gr_63.gif]
    
    Proof. (i): Since introducing groups _gr_64.gif] the right cancellation property says that introducing groups _gr_65.gif]
    (ii): Note introducing groups _gr_66.gif] and introducing groups _gr_67.gif] and thus introducing groups _gr_68.gif] since inverses are unique (or by using left cancellation).
    (iii): Note that introducing groups _gr_69.gif] and introducing groups _gr_70.gif] and so   introducing groups _gr_71.gif] since inverses are unique.
    (iv): If introducing groups _gr_72.gif] for all introducing groups _gr_73.gif] and since introducing groups _gr_74.gif] for all introducing groups _gr_75.gif] it follows that introducing groups _gr_76.gif] for all introducing groups _gr_77.gif] because inverses are unique. So, introducing groups _gr_78.gif] introducing groups _gr_79.gif] introducing groups _gr_80.gif] which proves that introducing groups _gr_81.gif] is Abelian.
    (v): If introducing groups _gr_82.gif] is Abelian, then introducing groups _gr_83.gif] Conversely, assume   introducing groups _gr_84.gif] for all introducing groups _gr_85.gif] Then introducing groups _gr_86.gif] introducing groups _gr_87.gif] introducing groups _gr_88.gif] introducing groups _gr_89.gif] introducing groups _gr_90.gif]

    Let introducing groups _gr_91.gif] be a group. Sometimes a group operation introducing groups _gr_92.gif] is denoted using standard multiplication notation; for example, introducing groups _gr_93.gif] might be denoted by introducing groups _gr_94.gif] or even introducing groups _gr_95.gif] The unique inverse of introducing groups _gr_96.gif] is denoted by introducing groups _gr_97.gif] The product introducing groups _gr_98.gif] is denoted by introducing groups _gr_99.gif] The   introducing groups _gr_100.gif] factors in the product introducing groups _gr_101.gif] is denoted by introducing groups _gr_102.gif] The identity of introducing groups _gr_103.gif] is sometimes denoted by introducing groups _gr_104.gif] and also by introducing groups _gr_105.gif] The notation introducing groups _gr_106.gif] means introducing groups _gr_107.gif] or introducing groups _gr_108.gif]
    Sometimes a group operation introducing groups _gr_109.gif] is denoted using standard addition notation (especially if the group is Abelian); for example, introducing groups _gr_110.gif] might be denoted by introducing groups _gr_111.gif] The unique inverse of introducing groups _gr_112.gif] is denoted by introducing groups _gr_113.gif] The sum introducing groups _gr_114.gif] is denoted by introducing groups _gr_115.gif] The introducing groups _gr_116.gif] terms in the sum introducing groups _gr_117.gif] is denoted by introducing groups _gr_118.gif] The identity of introducing groups _gr_119.gif] is sometimes denoted by introducing groups _gr_120.gif] and also by introducing groups _gr_121.gif] The notation introducing groups _gr_122.gif] means introducing groups _gr_123.gif] or introducing groups _gr_124.gif]

Proposition (Induction Properties) Let introducing groups _gr_125.gif] be a group with introducing groups _gr_126.gif]. Then

    (i) introducing groups _gr_127.gif] for all introducing groups _gr_128.gif]
    
    (ii) introducing groups _gr_129.gif] for all introducing groups _gr_130.gif] and
    
    (iii) introducing groups _gr_131.gif] for all introducing groups _gr_132.gif]
    
    Proof. (i): Let introducing groups _gr_133.gif] and prove that introducing groups _gr_134.gif] for all introducing groups _gr_135.gif] and all positive integers introducing groups _gr_136.gif] by induction. If introducing groups _gr_137.gif] then introducing groups _gr_138.gif] If introducing groups _gr_139.gif] then introducing groups _gr_140.gif] introducing groups _gr_141.gif] introducing groups _gr_142.gif] introducing groups _gr_143.gif] introducing groups _gr_144.gif] introducing groups _gr_145.gif] For introducing groups _gr_146.gif] introducing groups _gr_147.gif] introducing groups _gr_148.gif] introducing groups _gr_149.gif] introducing groups _gr_150.gif] If introducing groups _gr_151.gif] then introducing groups _gr_152.gif] introducing groups _gr_153.gif] introducing groups _gr_154.gif] introducing groups _gr_155.gif] introducing groups _gr_156.gif] introducing groups _gr_157.gif] as required.    
    Next prove that introducing groups _gr_158.gif] for any introducing groups _gr_159.gif] and for introducing groups _gr_160.gif] If introducing groups _gr_161.gif] for all introducing groups _gr_162.gif] then introducing groups _gr_163.gif] introducing groups _gr_164.gif] introducing groups _gr_165.gif] introducing groups _gr_166.gif] introducing groups _gr_167.gif] introducing groups _gr_168.gif] Now suppose introducing groups _gr_169.gif] for any introducing groups _gr_170.gif] Then introducing groups _gr_171.gif] introducing groups _gr_172.gif] introducing groups _gr_173.gif] introducing groups _gr_174.gif] Thus, introducing groups _gr_175.gif] for all introducing groups _gr_176.gif]  as required.    
    (ii): First, proof by induction on introducing groups _gr_177.gif] for introducing groups _gr_178.gif] Asume introducing groups _gr_179.gif] for any introducing groups _gr_180.gif] then introducing groups _gr_181.gif] introducing groups _gr_182.gif] introducing groups _gr_183.gif] introducing groups _gr_184.gif] introducing groups _gr_185.gif] and so introducing groups _gr_186.gif] holds for introducing groups _gr_187.gif] and introducing groups _gr_188.gif] Next, if introducing groups _gr_189.gif] and introducing groups _gr_190.gif] then introducing groups _gr_191.gif] introducing groups _gr_192.gif] introducing groups _gr_193.gif] introducing groups _gr_194.gif]  as required.    
    (iii): For any introducing groups _gr_195.gif] introducing groups _gr_196.gif]  as required. introducing groups _gr_197.gif]

Proposition (Group Characterization) If introducing groups _gr_198.gif] is a group and introducing groups _gr_199.gif] then the equations introducing groups _gr_200.gif] and introducing groups _gr_201.gif] have unique solutions. Conversely, if introducing groups _gr_202.gif] is a nonempty set with an associative binary operation in which the equations introducing groups _gr_203.gif] and introducing groups _gr_204.gif] have solutions for any introducing groups _gr_205.gif] then introducing groups _gr_206.gif] is a group.

    Proof.     The equations introducing groups _gr_207.gif] and introducing groups _gr_208.gif] have solutions, namely introducing groups _gr_209.gif] and introducing groups _gr_210.gif] respectively. Let introducing groups _gr_211.gif] be another solution for the equation introducing groups _gr_212.gif] Then multiplying both sides of the equation by introducing groups _gr_213.gif] yields introducing groups _gr_214.gif] and so introducing groups _gr_215.gif] Therefore introducing groups _gr_216.gif] is the only solution. Similarily, introducing groups _gr_217.gif] is the only solution to introducing groups _gr_218.gif]
     Conversely, assume the equations introducing groups _gr_219.gif] and introducing groups _gr_220.gif] have  solutions for all introducing groups _gr_221.gif] where introducing groups _gr_222.gif] is an associative operation on introducing groups _gr_223.gif] In particular, introducing groups _gr_224.gif] has a solution, say introducing groups _gr_225.gif] for some introducing groups _gr_226.gif] In fact introducing groups _gr_227.gif] is the identity of introducing groups _gr_228.gif] To see this let introducing groups _gr_229.gif] and let introducing groups _gr_230.gif] be any solution to the equation introducing groups _gr_231.gif] then introducing groups _gr_232.gif] introducing groups _gr_233.gif] introducing groups _gr_234.gif] introducing groups _gr_235.gif] Let introducing groups _gr_236.gif] introducing groups _gr_237.gif] is a solution to introducing groups _gr_238.gif] and introducing groups _gr_239.gif] be any solution to the equation introducing groups _gr_240.gif] then introducing groups _gr_241.gif] introducing groups _gr_242.gif] introducing groups _gr_243.gif] introducing groups _gr_244.gif] But then introducing groups _gr_245.gif] and introducing groups _gr_246.gif] and so introducing groups _gr_247.gif] Thus introducing groups _gr_248.gif] satisfies the requirements of being the identity element on introducing groups _gr_249.gif] Finally, given any introducing groups _gr_250.gif] let introducing groups _gr_251.gif] be a solution of the equation introducing groups _gr_252.gif] and let introducing groups _gr_253.gif] be a solution of the equation introducing groups _gr_254.gif] Then introducing groups _gr_255.gif] introducing groups _gr_256.gif] introducing groups _gr_257.gif] introducing groups _gr_258.gif] introducing groups _gr_259.gif] and so introducing groups _gr_260.gif] Thus introducing groups _gr_261.gif] and introducing groups _gr_262.gif] and so introducing groups _gr_263.gif] is an inverse for introducing groups _gr_264.gif] Therefore, introducing groups _gr_265.gif] is a group as required. introducing groups _gr_266.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 introducing groups _gr_267.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 introducing groups _gr_268.gif]

    Proof. The entries on the row for an element introducing groups _gr_269.gif] are introducing groups _gr_270.gif] introducing groups _gr_271.gif]..., introducing groups _gr_272.gif] Any element introducing groups _gr_273.gif] of introducing groups _gr_274.gif] appears as the entry introducing groups _gr_275.gif] with introducing groups _gr_276.gif] No element can appear more than once since that would mean that introducing groups _gr_277.gif] with introducing groups _gr_278.gif] but this is impossible because of left cancellation. The same applies for columns by the cancellation law. introducing groups _gr_279.gif]

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 introducing groups _gr_280.gif] be a set, then the set of all invertible mappings introducing groups _gr_281.gif] from introducing groups _gr_282.gif] to introducing groups _gr_283.gif] is a group with composition as the operation.
    
    (v) The set introducing groups _gr_284.gif] of all introducing groups _gr_285.gif] real matrices together with matrix addition as the operation is an Abelian group.
    
    (vi) The set introducing groups _gr_286.gif] of all introducing groups _gr_287.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   introducing groups _gr_288.gif] matrices over introducing groups _gr_289.gif]
    
    (vii) The set introducing groups _gr_290.gif] of all real-valued continuous functions on introducing groups _gr_291.gif] with pointwise addition (function addition) defined by introducing groups _gr_292.gif] is an Abelian group.  More generally, the set of all mappings from introducing groups _gr_293.gif] to introducing groups _gr_294.gif] with function addition is an Abelian group; and also the set
    
             introducing groups _gr_295.gif]    
                
is a group under function multiplication.

    (viii) The set introducing groups _gr_296.gif] consisiting of all subsets of a given set introducing groups _gr_297.gif] with operation defined by introducing groups _gr_298.gif] is an Abelian group.
    
    (ix) The set introducing groups _gr_299.gif] with complex multiplication is a group. In general, for introducing groups _gr_300.gif] the set introducing groups _gr_301.gif] of complex roots of unity
    
             introducing groups _gr_302.gif]
                
is an Abelian group under complex multiplication.

    (x) The set introducing groups _gr_303.gif] with componentwise addition is a group. In general, vector spaces are groups under vector addition.
    
    (xi) The set
    
introducing groups _gr_304.gif]

is an Abelian group under matrix multiplcation.

    (xii) The following matrices form a group under matrix multiplication and is called the quaternion group:
    
             introducing groups _gr_305.gif] introducing groups _gr_306.gif] introducing groups _gr_307.gif] introducing groups _gr_308.gif]
        
              introducing groups _gr_309.gif] introducing groups _gr_310.gif] introducing groups _gr_311.gif]     introducing groups _gr_312.gif]
introducing groups _gr_313.gif]

Cite this as:
Introducing Groups
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/introducing-groups.html
 
    
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