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Subgroup of a Group

    A subset of a group which is itself a group with respect to the same operation is called a subgroup. In this topic, the center and centralizer of a group are shown to be subgroups. Basically, the center of a group is the collection of elements in the group that commute with all elements in the group and the centralizer of a given element in the group is the collection of all elements in the group that commute with that given element. Of course, if the center is the whole group, then the group is Abelian; and the intersection of all the centralizers of the group is the center of the group. Many subgroup tests are detailed including the infamous “fastest gun” subgroup test.
    A subset subgroup of a group _gr_1.gif] of a group subgroup of a group _gr_2.gif] is a subgroup of subgroup of a group _gr_3.gif] if subgroup of a group _gr_4.gif] is itself a group with respect to the operation on subgroup of a group _gr_5.gif] There are two important subgroups any any group: the center of subgroup of a group _gr_6.gif] is the subset

subgroup of a group _gr_7.gif]

and the centralizer of subgroup of a group _gr_8.gif] in subgroup of a group _gr_9.gif] is the set

subgroup of a group _gr_10.gif]

Proposition (Subgroup of a Group) Let subgroup of a group _gr_11.gif] be a group with subgroup of a group _gr_12.gif]a nonempty subset of subgroup of a group _gr_13.gif]

    (i) If subgroup of a group _gr_14.gif] is a subgroup of subgroup of a group _gr_15.gif] subgroup of a group _gr_16.gif] is the identity of subgroup of a group _gr_17.gif] and subgroup of a group _gr_18.gif] is the identity of subgroup of a group _gr_19.gif] then subgroup of a group _gr_20.gif]
    
    (ii)  If subgroup of a group _gr_21.gif] is a subgroup of subgroup of a group _gr_22.gif] and subgroup of a group _gr_23.gif] then the inverse of subgroup of a group _gr_24.gif] in subgroup of a group _gr_25.gif] is the same as the inverse of subgroup of a group _gr_26.gif] in subgroup of a group _gr_27.gif]
    
    (iii) If subgroup of a group _gr_28.gif] and subgroup of a group _gr_29.gif] are subgroups of subgroup of a group _gr_30.gif] then subgroup of a group _gr_31.gif] is a subgroup of subgroup of a group _gr_32.gif]
    
    (iv) subgroup of a group _gr_33.gif] is a subgroup of subgroup of a group _gr_34.gif] if and only if subgroup of a group _gr_35.gif] is closed under   subgroup of a group _gr_36.gif] and every element of subgroup of a group _gr_37.gif] has an inverse in subgroup of a group _gr_38.gif]
    
    (v) (Fastest Gun) subgroup of a group _gr_39.gif] subgroup of a group _gr_40.gif] is a subgroup of subgroup of a group _gr_41.gif] if and only if subgroup of a group _gr_42.gif] for all subgroup of a group _gr_43.gif]
    
    (vi) subgroup of a group _gr_44.gif] is a subgroup of subgroup of a group _gr_45.gif]
    
    (vii)  If subgroup of a group _gr_46.gif] then subgroup of a group _gr_47.gif] is a subgroup of subgroup of a group _gr_48.gif] and subgroup of a group _gr_49.gif] is a subgroup of subgroup of a group _gr_50.gif]
    
    Proof.    (i): Let subgroup of a group _gr_51.gif] Then since subgroup of a group _gr_52.gif] is the identity on subgroup of a group _gr_53.gif] subgroup of a group _gr_54.gif] Since subgroup of a group _gr_55.gif] is the identity on subgroup of a group _gr_56.gif] subgroup of a group _gr_57.gif] Then by the cancellation law for subgroup of a group _gr_58.gif],   subgroup of a group _gr_59.gif] as desired.
    (ii): Suppose subgroup of a group _gr_60.gif] and that subgroup of a group _gr_61.gif] and subgroup of a group _gr_62.gif] are the inverses of subgroup of a group _gr_63.gif] in subgroup of a group _gr_64.gif] and subgroup of a group _gr_65.gif] repsectively. If subgroup of a group _gr_66.gif] is the identity, then subgroup of a group _gr_67.gif] By the cancellation law,   subgroup of a group _gr_68.gif] as required.
    (iii): Since subgroup of a group _gr_69.gif] is associative on subgroup of a group _gr_70.gif] subgroup of a group _gr_71.gif] is also associative on subgroup of a group _gr_72.gif] The identity on subgroup of a group _gr_73.gif] is in subgroup of a group _gr_74.gif] and subgroup of a group _gr_75.gif] and thus is in subgroup of a group _gr_76.gif] Let subgroup of a group _gr_77.gif] and subgroup of a group _gr_78.gif] be the inverse of subgroup of a group _gr_79.gif] in subgroup of a group _gr_80.gif] Then subgroup of a group _gr_81.gif] is in subgroup of a group _gr_82.gif] and subgroup of a group _gr_83.gif] and therefore subgroup of a group _gr_84.gif] is in subgroup of a group _gr_85.gif] By definition subgroup of a group _gr_86.gif] is a subgroup of subgroup of a group _gr_87.gif] as required.
    (iv): Assume that subgroup of a group _gr_88.gif] is closed under subgroup of a group _gr_89.gif] and that every element of subgroup of a group _gr_90.gif] has an inverse in subgroup of a group _gr_91.gif] Then since subgroup of a group _gr_92.gif] is closed on subgroup of a group _gr_93.gif] it is a binary operation on subgroup of a group _gr_94.gif] and since subgroup of a group _gr_95.gif] is associative on subgroup of a group _gr_96.gif] subgroup of a group _gr_97.gif] is also associative on subgroup of a group _gr_98.gif] If every element of subgroup of a group _gr_99.gif] has an inverse in subgroup of a group _gr_100.gif] then it remains to show that subgroup of a group _gr_101.gif]  (the identity) is actually in subgroup of a group _gr_102.gif] Let subgroup of a group _gr_103.gif] then there is a subgroup of a group _gr_104.gif] in subgroup of a group _gr_105.gif] such that subgroup of a group _gr_106.gif] But since subgroup of a group _gr_107.gif] is closed subgroup of a group _gr_108.gif] and so subgroup of a group _gr_109.gif] is a subgroup of subgroup of a group _gr_110.gif] Conversely, if subgroup of a group _gr_111.gif] is a subgroup of subgroup of a group _gr_112.gif] then, by definition of subgroup subgroup of a group _gr_113.gif] is closed under subgroup of a group _gr_114.gif] and every element of subgroup of a group _gr_115.gif] has an inverse in subgroup of a group _gr_116.gif]
    (v): If subgroup of a group _gr_117.gif] is a subgroup of subgroup of a group _gr_118.gif] then subgroup of a group _gr_119.gif] is closed under multiplication and all elements have inverses, thus subgroup of a group _gr_120.gif] for all subgroup of a group _gr_121.gif] Assume that for any subgroup of a group _gr_122.gif] subgroup of a group _gr_123.gif] Then subgroup of a group _gr_124.gif] because subgroup of a group _gr_125.gif] Also, since for any subgroup of a group _gr_126.gif] we have subgroup of a group _gr_127.gif] Finally, if subgroup of a group _gr_128.gif] then subgroup of a group _gr_129.gif] and so subgroup of a group _gr_130.gif] is a subgroup as desired.
    (vi): If subgroup of a group _gr_131.gif] then for any subgroup of a group _gr_132.gif] subgroup of a group _gr_133.gif] subgroup of a group _gr_134.gif] subgroup of a group _gr_135.gif] subgroup of a group _gr_136.gif] subgroup of a group _gr_137.gif] subgroup of a group _gr_138.gif] subgroup of a group _gr_139.gif] subgroup of a group _gr_140.gif] Therefore, subgroup of a group _gr_141.gif] and so subgroup of a group _gr_142.gif] is a subgroup of subgroup of a group _gr_143.gif] as desired.
    (vii): Let subgroup of a group _gr_144.gif] and to show that subgroup of a group _gr_145.gif] is a subgroup of subgroup of a group _gr_146.gif] let subgroup of a group _gr_147.gif] First note that subgroup of a group _gr_148.gif] subgroup of a group _gr_149.gif] subgroup of a group _gr_150.gif] subgroup of a group _gr_151.gif] Thus, subgroup of a group _gr_152.gif] subgroup of a group _gr_153.gif] subgroup of a group _gr_154.gif] subgroup of a group _gr_155.gif] subgroup of a group _gr_156.gif] subgroup of a group _gr_157.gif] subgroup of a group _gr_158.gif] subgroup of a group _gr_159.gif] Therefore, subgroup of a group _gr_160.gif] and so subgroup of a group _gr_161.gif] is a subgroup of subgroup of a group _gr_162.gif] as desired. If subgroup of a group _gr_163.gif] then subgroup of a group _gr_164.gif] for any given subgroup of a group _gr_165.gif] So subgroup of a group _gr_166.gif] and thus subgroup of a group _gr_167.gif] is a subgroup of subgroup of a group _gr_168.gif] subgroup of a group _gr_169.gif]

subgroup of a group _gr_170.gif] Recommended Links
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subgroup of a group _gr_183.gif] Recommended Math Books
subgroup of a group _gr_184.gif] Schaum's Outline of Group Theory
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subgroup of a group _gr_186.gif] Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups
subgroup of a group _gr_187.gif] The Subgroup Structure of the Finite Classical Groups (London Mathematical Society Lecture Note Series)
subgroup of a group _gr_188.gif] Algebraic Groups and Lie Groups with Few Factors (Lecture Notes in Mathematics)

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