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# U+22B2Normal Subgroup Of

U+22B2 was added in Unicode version 1.1 in 1993. It belongs to the block U+2200 to U+22FF Mathematical Operators in the U+0000 to U+FFFF Basic Multilingual Plane.

This character is a Math Symbol and is commonly used, that is, in no specific script.

The glyph is not a composition. It has no designated width in East Asian texts. In bidirectional text it acts as Other Neutral. When changing direction it is mirrored into Contains As Normal Subgroup. The word that U+22B2 forms with similar adjacent characters prevents a line break inside it. The glyph can be confused with one other glyph.

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup $N$ of the group $G$ is normal in $G$ if and only if $gn{g}^{-1}\in N$ for all $g\in G$ and $n\in N.$ The usual notation for this relation is $N◃G.$

Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of $G$ are precisely the kernels of group homomorphisms with domain $G,$ which means that they can be used to internally classify those homomorphisms.

Évariste Galois was the first to realize the importance of the existence of normal subgroups.

## Representations

System Representation
8882
UTF-8 E2 8A B2
UTF-16 22 B2
UTF-32 00 00 22 B2
URL-Quoted %E2%8A%B2
HTML hex reference &#x22B2;
Wrong windows-1252 Mojibake âŠ²
HTML named entity &LeftTriangle;
HTML named entity &vartriangleleft;
HTML named entity &vltri;
LATEX \vartriangleleft

## Complete Record

Property Value
Age (age) 1.1 (1993)
Unicode Name (na) NORMAL SUBGROUP OF
Unicode 1 Name (na1)
Block (blk) Mathematical Operators
General Category (gc) Math Symbol
Script (sc) Common
Bidirectional Category (bc) Other Neutral
Combining Class (ccc) Not Reordered
Decomposition Type (dt) none
Decomposition Mapping (dm) Normal Subgroup Of
Lowercase (Lower)
Simple Lowercase Mapping (slc) Normal Subgroup Of
Lowercase Mapping (lc) Normal Subgroup Of
Uppercase (Upper)
Simple Uppercase Mapping (suc) Normal Subgroup Of
Uppercase Mapping (uc) Normal Subgroup Of
Simple Titlecase Mapping (stc) Normal Subgroup Of
Titlecase Mapping (tc) Normal Subgroup Of
Case Folding (cf) Normal Subgroup Of
ASCII Hex Digit (AHex)
Alphabetic (Alpha)
Bidi Control (Bidi_C)
Bidi Mirrored (Bidi_M)
Composition Exclusion (CE)
Case Ignorable (CI)
Changes When Casefolded (CWCF)
Changes When Casemapped (CWCM)
Changes When NFKC Casefolded (CWKCF)
Changes When Lowercased (CWL)
Changes When Titlecased (CWT)
Changes When Uppercased (CWU)
Cased (Cased)
Full Composition Exclusion (Comp_Ex)
Default Ignorable Code Point (DI)
Dash (Dash)
Deprecated (Dep)
Diacritic (Dia)
Emoji Modifier Base (EBase)
Emoji Component (EComp)
Emoji Modifier (EMod)
Emoji Presentation (EPres)
Emoji (Emoji)
Extender (Ext)
Extended Pictographic (ExtPict)
FC NFKC Closure (FC_NFKC) Normal Subgroup Of
Grapheme Cluster Break (GCB) Any
Grapheme Base (Gr_Base)
Grapheme Extend (Gr_Ext)
Hex Digit (Hex)
Hyphen (Hyphen)
ID Continue (IDC)
ID Start (IDS)
IDS Binary Operator (IDSB)
IDS Trinary Operator and (IDST)
IDSU (IDSU) 0
ID_Compat_Math_Continue (ID_Compat_Math_Continue) 0
ID_Compat_Math_Start (ID_Compat_Math_Start) 0
Ideographic (Ideo)
InCB (InCB) None
Indic Mantra Category (InMC)
Indic Positional Category (InPC) NA
Indic Syllabic Category (InSC) Other
Jamo Short Name (JSN)
Join Control (Join_C)
Logical Order Exception (LOE)
Math (Math)
Noncharacter Code Point (NChar)
NFC Quick Check (NFC_QC) Yes
NFD Quick Check (NFD_QC) Yes
NFKC Casefold (NFKC_CF) Normal Subgroup Of
NFKC Quick Check (NFKC_QC) Yes
NFKC_SCF (NFKC_SCF) Normal Subgroup Of
NFKD Quick Check (NFKD_QC) Yes
Other Alphabetic (OAlpha)
Other Default Ignorable Code Point (ODI)
Other Grapheme Extend (OGr_Ext)
Other ID Continue (OIDC)
Other ID Start (OIDS)
Other Lowercase (OLower)
Other Math (OMath)
Other Uppercase (OUpper)
Prepended Concatenation Mark (PCM)
Pattern Syntax (Pat_Syn)
Pattern White Space (Pat_WS)
Quotation Mark (QMark)
Regional Indicator (RI)
Sentence Break (SB) Other
Soft Dotted (SD)
Sentence Terminal (STerm)
Terminal Punctuation (Term)
Unified Ideograph (UIdeo)
Variation Selector (VS)
Word Break (WB) Other
White Space (WSpace)
XID Continue (XIDC)
XID Start (XIDS)
Expands On NFC (XO_NFC)
Expands On NFD (XO_NFD)
Expands On NFKC (XO_NFKC)
Expands On NFKD (XO_NFKD)
Bidi Mirrored Glyph (bmg) Contains As Normal Subgroup
Bidi Paired Bracket (bpb) Normal Subgroup Of
Bidi Paired Bracket Type (bpt) None
East Asian Width (ea) neutral
Hangul Syllable Type (hst) Not Applicable
ISO 10646 Comment (isc)
Joining Group (jg) No_Joining_Group
Joining Type (jt) Non Joining
Line Break (lb) Alphabetic
Numeric Type (nt) none
Numeric Value (nv) not a number
Simple Case Folding (scf) Normal Subgroup Of
Script Extension (scx)
Vertical Orientation (vo) R