Introducing Cosets of a Subgroup

    This topic defines two equivalence relations on any group whose equivalence classes are called right cosets and left cosets, respectively. If a group is Abelian then it turns out that right cosets and left cosets are the same. Examples are given which demonstrate how to compute the cosets of a subgroup of a finite group and that differentiating between right cosets and left cosets is significant. This topic also states the famous Lagrange’s Theorem and gives some of its consequences. Basically Lagrange’s Theorem states that the order of a finite subgroup is a divisor of the order of the group.

Proposition (Right Cosets) If introducing cosets of a subgroup _gr_1.gif] is a subgroup of introducing cosets of a subgroup _gr_2.gif], then the relation introducing cosets of a subgroup _gr_3.gif] defined on introducing cosets of a subgroup _gr_4.gif] by introducing cosets of a subgroup _gr_5.gif] if and only if introducing cosets of a subgroup _gr_6.gif] is an equivalence relation of introducing cosets of a subgroup _gr_7.gif]

    Proof. If introducing cosets of a subgroup _gr_8.gif] then introducing cosets of a subgroup _gr_9.gif] because introducing cosets of a subgroup _gr_10.gif] If introducing cosets of a subgroup _gr_11.gif] then introducing cosets of a subgroup _gr_12.gif] and so introducing cosets of a subgroup _gr_13.gif] implies introducing cosets of a subgroup _gr_14.gif] If introducing cosets of a subgroup _gr_15.gif] and introducing cosets of a subgroup _gr_16.gif] then introducing cosets of a subgroup _gr_17.gif] and introducing cosets of a subgroup _gr_18.gif] and so introducing cosets of a subgroup _gr_19.gif] Thus, introducing cosets of a subgroup _gr_20.gif] is reflexive, symmetric, and transitive as desired. introducing cosets of a subgroup _gr_21.gif]

Definition (Right Cosets) If introducing cosets of a subgroup _gr_22.gif] is a subgroup of introducing cosets of a subgroup _gr_23.gif], then the set of equivalence classes for the equivalence relation defined by introducing cosets of a subgroup _gr_24.gif] if and only if introducing cosets of a subgroup _gr_25.gif] are called the right cosets of introducing cosets of a subgroup _gr_26.gif] in introducing cosets of a subgroup _gr_27.gif]

    The right coset of introducing cosets of a subgroup _gr_28.gif] to which introducing cosets of a subgroup _gr_29.gif] belongs is denoted introducing cosets of a subgroup _gr_30.gif] Indeed, if the group operation is introducing cosets of a subgroup _gr_31.gif] then cosets are denoted by introducing cosets of a subgroup _gr_32.gif] In particular,  if the operation is addition or multiplication, then we denote cosets by introducing cosets of a subgroup _gr_33.gif] or introducing cosets of a subgroup _gr_34.gif] respectively.

Example (Coset Notation) Let introducing cosets of a subgroup _gr_35.gif] and introducing cosets of a subgroup _gr_36.gif] Then

introducing cosets of a subgroup _gr_37.gif]

which is the equivalence class introducing cosets of a subgroup _gr_38.gif] in introducing cosets of a subgroup _gr_39.gif] introducing cosets of a subgroup _gr_40.gif]

Proposition (Cosets of a Subgroup) If introducing cosets of a subgroup _gr_41.gif] is a subgroup of introducing cosets of a subgroup _gr_42.gif] and introducing cosets of a subgroup _gr_43.gif] then the following are equivalent conditions:

    (i) introducing cosets of a subgroup _gr_44.gif]
    
    (ii) introducing cosets of a subgroup _gr_45.gif] for some introducing cosets of a subgroup _gr_46.gif]
    
    (iii) introducing cosets of a subgroup _gr_47.gif]
    
    (iv) introducing cosets of a subgroup _gr_48.gif]
    
Moreover, if introducing cosets of a subgroup _gr_49.gif] is finite, then introducing cosets of a subgroup _gr_50.gif]

    Proof. introducing cosets of a subgroup _gr_51.gif] If introducing cosets of a subgroup _gr_52.gif] then introducing cosets of a subgroup _gr_53.gif] for some introducing cosets of a subgroup _gr_54.gif] and so introducing cosets of a subgroup _gr_55.gif] for some introducing cosets of a subgroup _gr_56.gif]
     introducing cosets of a subgroup _gr_57.gif] If   introducing cosets of a subgroup _gr_58.gif] for some introducing cosets of a subgroup _gr_59.gif] then introducing cosets of a subgroup _gr_60.gif] and so introducing cosets of a subgroup _gr_61.gif] which means that introducing cosets of a subgroup _gr_62.gif] is the right coset of introducing cosets of a subgroup _gr_63.gif] that is introducing cosets of a subgroup _gr_64.gif]
     introducing cosets of a subgroup _gr_65.gif] Suppose introducing cosets of a subgroup _gr_66.gif] or equivalently introducing cosets of a subgroup _gr_67.gif]. If introducing cosets of a subgroup _gr_68.gif] then introducing cosets of a subgroup _gr_69.gif] and so introducing cosets of a subgroup _gr_70.gif] by transitivity. Therefore, introducing cosets of a subgroup _gr_71.gif] Conversely, follows by symmetry.
     introducing cosets of a subgroup _gr_72.gif] Given introducing cosets of a subgroup _gr_73.gif] then introducing cosets of a subgroup _gr_74.gif] is in the right coset of introducing cosets of a subgroup _gr_75.gif] and conversely. Thus, introducing cosets of a subgroup _gr_76.gif] which means introducing cosets of a subgroup _gr_77.gif]  
    Suppose introducing cosets of a subgroup _gr_78.gif] is finite and define the mapping introducing cosets of a subgroup _gr_79.gif] by introducing cosets of a subgroup _gr_80.gif] for each introducing cosets of a subgroup _gr_81.gif] This mapping is well-defined by the operation on introducing cosets of a subgroup _gr_82.gif] Further, introducing cosets of a subgroup _gr_83.gif] is one-to-one because if introducing cosets of a subgroup _gr_84.gif] then introducing cosets of a subgroup _gr_85.gif] and the right cancellation property, implies introducing cosets of a subgroup _gr_86.gif] Also, introducing cosets of a subgroup _gr_87.gif] is onto since introducing cosets of a subgroup _gr_88.gif] consists of elements of the form introducing cosets of a subgroup _gr_89.gif] for some introducing cosets of a subgroup _gr_90.gif] by parts (i)-(iv). introducing cosets of a subgroup _gr_91.gif]

     One right coset of introducing cosets of a subgroup _gr_92.gif] in introducing cosets of a subgroup _gr_93.gif] will be introducing cosets of a subgroup _gr_94.gif] To compute all the right cosets of introducing cosets of a subgroup _gr_95.gif] in a finite group introducing cosets of a subgroup _gr_96.gif] first choose any element in the complement of introducing cosets of a subgroup _gr_97.gif] and compute introducing cosets of a subgroup _gr_98.gif] Then choose any element introducing cosets of a subgroup _gr_99.gif] and compute introducing cosets of a subgroup _gr_100.gif]If we continue until introducing cosets of a subgroup _gr_101.gif] is exhausted we have computed all right cosets of introducing cosets of a subgroup _gr_102.gif]  

Example (Right Cosets) Let introducing cosets of a subgroup _gr_103.gif] and introducing cosets of a subgroup _gr_104.gif] Then,
introducing cosets of a subgroup _gr_105.gif]
are the right cosets of introducing cosets of a subgroup _gr_106.gif]It is important to notice that these right cosets partition the group. introducing cosets of a subgroup _gr_107.gif]

Example (Right Cosets) Let introducing cosets of a subgroup _gr_108.gif] be the subgroup generated by introducing cosets of a subgroup _gr_109.gif] in introducing cosets of a subgroup _gr_110.gif]The first coset of notice is introducing cosets of a subgroup _gr_111.gif] Then by choosing an element not in introducing cosets of a subgroup _gr_112.gif] say introducing cosets of a subgroup _gr_113.gif] we form the products introducing cosets of a subgroup _gr_114.gif] and introducing cosets of a subgroup _gr_115.gif] to obtain the coset introducing cosets of a subgroup _gr_116.gif] Next we choose any element not in either of the first two cosets, say introducing cosets of a subgroup _gr_117.gif] which gives introducing cosets of a subgroup _gr_118.gif] Continuing in this fashion we obtain introducing cosets of a subgroup _gr_119.gif] introducing cosets of a subgroup _gr_120.gif] introducing cosets of a subgroup _gr_121.gif] introducing cosets of a subgroup _gr_122.gif] and introducing cosets of a subgroup _gr_123.gif] introducing cosets of a subgroup _gr_124.gif]

    Similiarily to right cosets we can define left cosets as follows: if introducing cosets of a subgroup _gr_125.gif] is a subgroup of introducing cosets of a subgroup _gr_126.gif], then the set of equivalence classes for the equivalence relation defined by introducing cosets of a subgroup _gr_127.gif] if and only if introducing cosets of a subgroup _gr_128.gif] are called the left cosets of introducing cosets of a subgroup _gr_129.gif] in introducing cosets of a subgroup _gr_130.gif] For non-Abelian groups left and right cosets must be distinguished.

Example (Distinguish Between Left and Right Cosets) Let introducing cosets of a subgroup _gr_131.gif] be the group of all permutations on a set with three elements where introducing cosets of a subgroup _gr_132.gif] introducing cosets of a subgroup _gr_133.gif] and introducing cosets of a subgroup _gr_134.gif] and let introducing cosets of a subgroup _gr_135.gif] Then the left cosets of introducing cosets of a subgroup _gr_136.gif] are introducing cosets of a subgroup _gr_137.gif] introducing cosets of a subgroup _gr_138.gif] and introducing cosets of a subgroup _gr_139.gif] and the right cosets of introducing cosets of a subgroup _gr_140.gif] are introducing cosets of a subgroup _gr_141.gif] introducing cosets of a subgroup _gr_142.gif] introducing cosets of a subgroup _gr_143.gif] introducing cosets of a subgroup _gr_144.gif]

Proposition (Lagrange's Theorem) If introducing cosets of a subgroup _gr_145.gif] is a subgroup of a finite group introducing cosets of a subgroup _gr_146.gif] then the order of introducing cosets of a subgroup _gr_147.gif] is a divisor of the order of introducing cosets of a subgroup _gr_148.gif]

    Proof. Given a subgroup introducing cosets of a subgroup _gr_149.gif] we can form the set of all right cosets of introducing cosets of a subgroup _gr_150.gif] say there are introducing cosets of a subgroup _gr_151.gif] of them, and we can choose one element from each coset say introducing cosets of a subgroup _gr_152.gif] Then introducing cosets of a subgroup _gr_153.gif] since all the right cosets partition introducing cosets of a subgroup _gr_154.gif] Since introducing cosets of a subgroup _gr_155.gif] and introducing cosets of a subgroup _gr_156.gif] for any introducing cosets of a subgroup _gr_157.gif] it follow that introducing cosets of a subgroup _gr_158.gif] Thus,  the order of introducing cosets of a subgroup _gr_159.gif] is a divisor of the order of introducing cosets of a subgroup _gr_160.gif] as desired. introducing cosets of a subgroup _gr_161.gif]

Proposition (Consequences of the Lagrange Theorem)

    (i) If introducing cosets of a subgroup _gr_162.gif] is a finite group with introducing cosets of a subgroup _gr_163.gif] then introducing cosets of a subgroup _gr_164.gif]
    
    (ii) Any group introducing cosets of a subgroup _gr_165.gif] of prime order contains no subgroup other than introducing cosets of a subgroup _gr_166.gif] and introducing cosets of a subgroup _gr_167.gif] is cyclic, and is generated by one of its nonidentity elements.
    
    Proof. (i) Since introducing cosets of a subgroup _gr_168.gif] and introducing cosets of a subgroup _gr_169.gif] is a subgroup introducing cosets of a subgroup _gr_170.gif] Therefore, introducing cosets of a subgroup _gr_171.gif] introducing cosets of a subgroup _gr_172.gif] introducing cosets of a subgroup _gr_173.gif] introducing cosets of a subgroup _gr_174.gif] introducing cosets of a subgroup _gr_175.gif] introducing cosets of a subgroup _gr_176.gif] for some integer introducing cosets of a subgroup _gr_177.gif]
    (ii) Since a prime only has 1 and itself as a divisor there are no subgroups of introducing cosets of a subgroup _gr_178.gif] besides introducing cosets of a subgroup _gr_179.gif] and introducing cosets of a subgroup _gr_180.gif] If introducing cosets of a subgroup _gr_181.gif] then introducing cosets of a subgroup _gr_182.gif] and so introducing cosets of a subgroup _gr_183.gif] is cyclic as desired. introducing cosets of a subgroup _gr_184.gif]

Cite this as:
Introducing Cosets Of A Subgroup
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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