Home Metamath Proof ExplorerTheorem List (p. 150 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubgid 14901 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
SubGrp

Theoremsubggrp 14902 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
s        SubGrp

Theoremsubgbas 14903 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
s        SubGrp

Theoremsubgrcl 14904 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubg0 14905 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
s               SubGrp

Theoremsubginv 14906 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
s                      SubGrp

Theoremsubg0cl 14907 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubginvcl 14908 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgcl 14909 A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgsubcl 14910 A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremsubgsub 14911 The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               SubGrp

Theoremsubgmulgcl 14912 Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
.g       SubGrp

Theoremsubgmulg 14913 A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
.g       s        .g       SubGrp

Theoremissubg2 14914* Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremissubg3 14915* A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubGrp SubMnd

Theoremissubg4 14916* A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp

Theoremsubgsubm 14917 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
SubGrp SubMnd

Theoremsubsubg 14918 A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
s        SubGrp SubGrp SubGrp

Theoremsubgint 14919 The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
SubGrp SubGrp

Theorem0subg 14920 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
SubGrp

Theoremcycsubgcl 14921* The set of integer powers of an element of a group forms a subgroup containing , called the cyclic group generated by the element . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremcycsubgss 14922* The cyclic subgroup generated by an element is a subset of any subgroup containing . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremcycsubg 14923* The cyclic group generated by is the smallest subgroup containing . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremisnsg 14924* Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremisnsg2 14925* Weaken the condition of isnsg 14924 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnsgbi 14926 Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp

Theoremnsgsubg 14927 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnsgconj 14928 The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremisnsg3 14929* A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremsubgacs 14930 Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
SubGrp ACS

Theoremnsgacs 14931 Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
NrmSGrp ACS

Theoremcycsubg2 14932* The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g              mrClsSubGrp

Theoremcycsubg2cl 14933 Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
.g       mrClsSubGrp

Theoremelnmz 14934* Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Theoremnmzbi 14935* Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Theoremnmzsubg 14936* The normalizer NG(S) of a subset of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremssnmz 14937* A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremisnsg4 14938* A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnmznsg 14939* Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
s        SubGrp NrmSGrp

Theorem0nsg 14940 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremnsgid 14941 The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremreleqg 14942 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremeqgfval 14943* Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
~QG

Theoremeqgval 14944 Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremeqger 14945 The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
~QG        SubGrp

Theoremeqglact 14946* A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG

Theoremeqgid 14947 The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremeqgen 14948 Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG        SubGrp

Theoremeqgcpbl 14949 The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG               NrmSGrp

Theoremdivsgrp 14950 If is a normal subgroup of , then is a group, called the quotient of by . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
s ~QG        NrmSGrp

Theoremdivseccl 14951 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                      NrmSGrp ~QG

Theoremdivsadd 14952 Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG ~QG

Theoremdivs0 14953 Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG               NrmSGrp ~QG

Theoremdivsinv 14954 Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG

Theoremdivssub 14955 Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG ~QG

Theoremlagsubg2 14956 Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
~QG        SubGrp

Theoremlagsubg 14957 Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
SubGrp

10.2.3  Elementary theory of group homomorphisms

Syntaxcghm 14958 Extend class notation with the generator of group hom-sets.

Definitiondf-ghm 14959* A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremreldmghm 14960 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghm 14961* Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghm3 14962* Property of a group homomorphism, similar to ismhm 14695. (Contributed by Mario Carneiro, 7-Mar-2015.)

Theoremghmgrp1 14963 A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmgrp2 14964 A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmf 14965 A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmlin 14966 A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmid 14967 A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghminv 14968 A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmsub 14969 Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghmd 14970* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)

Theoremghmmhm 14971 A group homorphism is a monoid homorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
MndHom

Theoremghmmhmb 14972 Group homorphisms and monoid homomorphisms coincide. (Thus, is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
MndHom

Theoremghmmulg 14973 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g       .g

Theoremghmrn 14974 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp

Theorem0ghm 14975 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremidghm 14976 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremresghm 14977 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
s        SubGrp

Theoremresghm2 14978 One direction of resghm2b 14979. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
s        SubGrp

Theoremresghm2b 14979 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
s        SubGrp

Theoremghmco 14980 The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)

Theoremghmima 14981 The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp SubGrp

Theoremghmpreima 14982 The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp SubGrp

Theoremghmeql 14983 The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
SubGrp

Theoremghmnsgima 14984 The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp NrmSGrp

Theoremghmnsgpreima 14985 The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp NrmSGrp

Theoremghmker 14986 The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremghmeqker 14987 Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theorempwsdiagghm 14988* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s

Theoremghmf1 14989* Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremghmf1o 14990 A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)

Theoremconjghm 14991* Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)

Theoremconjsubg 14992* A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
SubGrp SubGrp

Theoremconjsubgen 14993* A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremconjnmz 14994* A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremconjnmzb 14995* Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremconjnsg 14996* A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp

Theoremdivsghm 14997* If is a normal subgroup of , then the "natural map" from elements to their cosets is a group homomorphism from to . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
s ~QG        ~QG        NrmSGrp

Theoremghmpropd 14998* Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)

10.2.4  Isomorphisms of groups

Syntaxcgim 14999 The class of group isomorphism sets.
GrpIso

Syntaxcgic 15000 The class of the group isomorphism relation.
𝑔

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
 Copyright terms: Public domain < Previous  Next >