If you are given a set of elements and a binary operation defined on pairs of elements in the set, can you determine some properties of this set and operation? Can you determine if the set of elements and associated operation satisfies the conditions for being an algebraic group? <\/p>\n
A group is an algebraic entity with a set G and a binary operation (an operation that involves two elements from the set), often denoted *, that satisfies the following conditions: <\/p>\n
Additionally we will say a group G is commutative if for all elements a and b in G, a b = b <\/em> a and a * b is also an element in G. <\/p>\n We define the n-th power of some element a in G to be a^n where n is a positive integer and a^(n+1) is defined recursively as follows: <\/p>\n a^(n+1) = a * a^n <\/p>\n a^1 = a <\/p>\n We also define the order of an element a in G to be the smallest possible positive integer m such that a^m = e, where e is the identity of the group. Note that order is only well defined in a group. <\/p>\n Formally, a non-empty subset S of a group G is a subgroup of G if:<\/p>\n In this lab, the elements in the sets are all Strings. These strings are merely symbols, which are either equal or distinct: they have no deeper meaning or significance in terms of their contents. <\/p>\n A binary operation on these String elements is a mapping that takes in an ordered pair of String elements and assigns it to another String element, which may be equal or distinct from either String operand. <\/p>\n Let G = { "x1", "x2", "x3" } be a set of Strings, and take * to be defined by the operation table below. <\/p>\n * x1 x2 x3 <\/p>\n x1 x2 x3 x1 <\/p>\n x2 x3 x1 x2<\/p>\n x3 x1 x2 x3<\/p>\n The table above looks like a multiplication table, except that the x values need not be integers and the operation need not be multiplication. <\/p>\n This operation can be equivalently written using the pair of operands (inputs to the operation) and the output as follows: <\/p>\n It is essential to recognize that this structure of mapping an ordered pair of input elements defines the operation * for G . <\/p>\n Consider for a moment the binary operation addition + defined on the integers. Since you know how to add, you have a procedure in your mind for how to determine the output of any addition operation. However, you could also view the addition operation for the integers as being defined by an (infinitely) large operation table of the form we have presented here, where each row corresponds to a mapping of two inputs to their respective output. When we can’t easily reduce a binary operation over a finite number of elements into an easy-to-explain procedure, we can represent it instead by an operation table that enumerates all input-output pairs defined by the operation. This representation of a binary operation using an operation table always exists whether or not we choose to use it to reason about the operation or not. For operations defined on finite sets (like those in this lab), the tabular representation is more useful. <\/p>\n<\/blockquote>\n Now, let’s return to the set G = { "x1", "x2", "x3" } defined by the operation table in the previous section. Does this set and operation pair represent a group? We see that all four conditions of a group hold, therefore indeed a group. G and * are indeed a Group.<\/p>\n and 3 is the smallest possible positive integer such that x1^3 =e<\/p>\n The ProbableGroup type, which you will implement, is created using a set of elements and an operation on pairs of elements (this operation is described explicitly using a table that maps a pair of values to the output value). <\/p>\n In Java vocabulary, the ProbableGroup<\/strong> class has a constructor public ProbableGroup(Set elements, Map, String>) opTable). <\/p>\n Relating to the background<\/strong> section, elements is our set G and opTable<\/strong> is our operation table. opTable<\/strong> is initialized such that opTable.get(new Pair("x1","x2"))<\/strong> returns the result of the operation x1 * x2, and x1, x2 are included in elements.<\/p>\n You need to implement the constructor and complete the following methods: <\/p>\n More detailed specs provided in the skeleton code and we recommend that you implement methods in the provided order. <\/p>\n You can assume any constraint stated under @param<\/strong> or requires<\/strong> in the specs is always satisfied. (You do not need to worry about how to handle violations of the preconditions.) <\/p>\n Accept the "Lab 4: Abstract Algebra" assignment via the Github Classroom link to create a new personal repo for your solution. <\/p>\n<\/li>\n Clone your personalized Implement the ProbableGroup class in Test your implementation using the provided smoke tests and any other tests you write to build confidence in the robustness of your implementation. <\/p>\n<\/li>\n (Optional). Copy and paste the contents of your ProbableGroup.java file into the code editor for question LB4.1 on PrairieLearn and submit your work.<\/p>\n Push your work to GitHub. <\/p>\n<\/li>\n Check GitHub to ensure your latest commit is visible.<\/p>\n<\/li>\n<\/ol>\n Here\u2019s a clean, immutable setup with the constructor and first few methods:<\/p>\n java<\/p>\n Here\u2019s a simple test you can run to check java<\/p>\n This example assumes Let\u2019s tackle both the This method checks whether a given subset java<\/p>\n This ensures:<\/p>\n Associativity is the trickiest group axiom to verify because it requires checking all combinations of three elements:<\/p>\n java<\/p>\n This method also checks:<\/p>\n Here\u2019s a simple test you can run in java<\/p>\n This setup assumes associativity holds (you\u2019ll want to verify manually or with a truth table). If you want to test a non-group, just break one of the axioms\u2014like removing an inverse or altering a product.<\/p>\n \u9019\u6bb5 Java \u7a0b\u5f0f\u78bc\u5b9a\u7fa9\u4e86\u4e00\u500b\u4e0d\u53ef\u8b8a\u7684\u985e\u5225 java<\/p>\n \u8907\u88fd<\/p>\n java<\/p>\n \u8907\u88fd<\/p>\n\n
Background: Binary Operations Using Operation Tables<\/h3>\n
\n\n
\n \nxi, xj<\/th>\n xi * xj<\/th>\n<\/tr>\n<\/thead>\n \n operand pair<\/td>\n output<\/td>\n<\/tr>\n \n x1, x1<\/td>\n x2<\/td>\n<\/tr>\n \n x1, x2<\/td>\n x3<\/td>\n<\/tr>\n \n x1, x3<\/td>\n x1<\/td>\n<\/tr>\n \n x2, x1<\/td>\n x3<\/td>\n<\/tr>\n \n x2, x2<\/td>\n x1<\/td>\n<\/tr>\n \n x2, x3<\/td>\n x2<\/td>\n<\/tr>\n \n x3, x1<\/td>\n x1<\/td>\n<\/tr>\n \n x3, x2<\/td>\n x2<\/td>\n<\/tr>\n \n x3, x3<\/td>\n x3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n
An Example<\/h3>\n
\nIs the operation commutative? What is the order of x1 ?
\nTo answer the questions posed at the start of this example, we need to check and see if G and * satisfy the four conditions we mention in the beginning. <\/p>\n\n
x1 * x3 = x3 * x1 = x1\nx2 * x3 = x3 * x2 = x2 x3 * \nx3 = x3 x3 <\/code><\/pre>\n2. All elements of G have their inverse as an element of G. We know the x3 is identity, and so for any xi, the inverse is some xj , the inverse of that xi is some xj such that xi * xj = xj * xi = x3. In our example we see<\/code><\/pre>\nx1 * x2 = x2 * x1 = x3 = e \nx2 * x1 = x1 * x2 = x3 = e \nx3 * x3 = x3 = e <\/code><\/pre>\n\n
x1 * (x2 * x3) = x1 * x2 = x3\n(x1 * x2) * x3 = x3 * x3 = x3\n=> x1 * (x2 * x3) = (x1 * x2) * x3 <\/code><\/pre>\n\n
\n
\nWe know this to be true as the product table is<\/li>\nx1^3 = x1 * x1^2 = x1 * (x1 * x1) = x1 * x2 = x3 = e<\/code><\/pre>\n\n
The ProbableGroup Type<\/h3>\n
\n
Task <\/code><\/pre>\nThere are 2 PrairieLearn assessments associated with this lab. Assessment LB4.1 is a zero-credit assessment for you to test your implementation for the programming exercise. Assessment LB4.2 corresponds to the conceptual questions for this lab. For the programming exercise in this lab, only work submitted via GitHub will be graded. <\/code><\/pre>\nProgramming Exercise (via GitHub) <\/code><\/pre>\n\n
lab-4-*<\/code> repo onto your computer and open it as an IntelliJ project.<\/p>\n<\/li>\n
\n…\/src\/main\/java\/groups\/ProbableGroup.java <\/p>\n\n
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\ud83e\udde0 Class Design Strategy<\/h3>\n
\n
\n
Set<String> elements<\/code>: the group set GG<\/li>\nMap<Pair<String, String>, String> opTable<\/code>: the binary operation \u2217*<\/li>\n<\/ul>\n<\/li>\n\n
Collections.unmodifiableSet<\/code> and Collections.unmodifiableMap<\/code> in the constructor.<\/li>\nfinal<\/code>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\ud83d\udd27 Method Implementation Guide<\/h3>\n
1.
product(String a, String b)<\/code><\/h4>\n\n
opTable.get(new Pair<>(a, b))<\/code><\/li>\n<\/ul>\n2.
power(String a, int n)<\/code><\/h4>\n\n
product<\/code>, starting from identity (or a<\/code> if n=1n = 1)<\/li>\n3.
getIdentity()<\/code><\/h4>\n\n
e<\/code> in elements<\/code> such that for all x<\/code>, product(e, x) == x<\/code> and product(x, e) == x<\/code><\/li>\n<\/ul>\n4.
getOrder(String a)<\/code><\/h4>\n\n
a<\/code> until you reach identity<\/li>\n5.
getInverse(String a)<\/code><\/h4>\n\n
b<\/code> in elements<\/code> such that product(a, b) == e<\/code> and product(b, a) == e<\/code><\/li>\n<\/ul>\n6.
isGroup()<\/code><\/h4>\n\n
\n
opTable<\/code> are in elements<\/code><\/li>\n7.
isCommutative()<\/code><\/h4>\n\n
8. Immutability<\/strong><\/h4>\n
\n
elements<\/code> and opTable<\/code> with unmodifiable wrappers<\/li>\n9.
isSubGroup(Set<String> h)<\/code><\/h4>\n\n
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\ud83e\uddea Testing Tips<\/h3>\n
\n
\n
\ud83e\uddf1 Starter Skeleton:
ProbableGroup.java<\/code><\/h3>\npackage groups;\n\nimport java.util.*;\n\npublic class ProbableGroup {\n private final Set<String> elements;\n private final Map<Pair<String, String>, String> opTable;\n\n public ProbableGroup(Set<String> elements, Map<Pair<String, String>, String> opTable) {\n this.elements = Collections.unmodifiableSet(new HashSet<>(elements));\n this.opTable = Collections.unmodifiableMap(new HashMap<>(opTable));\n }\n\n public String product(String a, String b) {\n return opTable.get(new Pair<>(a, b));\n }\n\n public String power(String a, int n) {\n if (n == 0) return getIdentity();\n String result = a;\n int absN = Math.abs(n);\n for (int i = 1; i < absN; i++) {\n result = product(result, a);\n }\n if (n < 0) result = power(getInverse(a), absN);\n return result;\n }\n\n public String getIdentity() {\n for (String e : elements) {\n boolean isIdentity = true;\n for (String x : elements) {\n if (!product(e, x).equals(x) || !product(x, e).equals(x)) {\n isIdentity = false;\n break;\n }\n }\n if (isIdentity) return e;\n }\n return null;\n }\n\n public String getInverse(String a) {\n String identity = getIdentity();\n for (String b : elements) {\n if (product(a, b).equals(identity) && product(b, a).equals(identity)) {\n return b;\n }\n }\n return null;\n }\n\n \/\/ Add other methods here...\n}<\/code><\/pre>\n\u2705 Sample Test Case: Subgroup Validation<\/h3>\n
isSubGroup(Set<String> h)<\/code> once implemented:<\/p>\npublic static void main(String[] args) {\n Set<String> elements = Set.of("e", "a", "b");\n Map<Pair<String, String>, String> opTable = Map.of(\n new Pair<>("e", "e"), "e",\n new Pair<>("e", "a"), "a",\n new Pair<>("e", "b"), "b",\n new Pair<>("a", "e"), "a",\n new Pair<>("a", "a"), "e",\n new Pair<>("a", "b"), "b",\n new Pair<>("b", "e"), "b",\n new Pair<>("b", "a"), "b",\n new Pair<>("b", "b"), "e"\n );\n\n ProbableGroup group = new ProbableGroup(elements, opTable);\n Set<String> h = Set.of("e", "b");\n\n System.out.println("Is subgroup: " + group.isSubGroup(h)); \/\/ Expected: true\n}<\/code><\/pre>\nb * b = e<\/code>, b * e = b<\/code>, and e<\/code> is the identity\u2014so e, b<\/code> forms a valid subgroup.<\/p>\nisSubGroup<\/code> method and a test for associativity. These will help you validate group structure and deepen your understanding of abstract algebra in code.<\/p>\n\ud83e\uddee
isSubGroup<\/code> Implementation<\/h3>\nh<\/code> satisfies the subgroup criteria:<\/p>\npublic boolean isSubGroup(Set<String> h) {\n \/\/ Must be a subset of the group\n if (!elements.containsAll(h)) return false;\n\n String identity = getIdentity();\n if (identity == null || !h.contains(identity)) return false;\n\n for (String a : h) {\n String inverse = getInverse(a);\n if (inverse == null || !h.contains(inverse)) return false;\n\n for (String b : h) {\n String prod = product(a, b);\n if (prod == null || !h.contains(prod)) return false;\n }\n }\n return true;\n}<\/code><\/pre>\n\n
\ud83d\udd01 Associativity Test<\/h3>\n
public boolean isGroup() {\n String identity = getIdentity();\n if (identity == null) return false;\n\n for (String a : elements) {\n String inverse = getInverse(a);\n if (inverse == null) return false;\n\n for (String b : elements) {\n for (String c : elements) {\n String left = product(product(a, b), c);\n String right = product(a, product(b, c));\n if (!left.equals(right)) return false;\n }\n }\n }\n return true;\n}<\/code><\/pre>\n\n
\ud83e\uddea Test Case for Associativity<\/h3>\n
main()<\/code>:<\/p>\nSet<String> elements = Set.of("e", "a", "b");\nMap<Pair<String, String>, String> opTable = Map.of(\n new Pair<>("e", "e"), "e",\n new Pair<>("e", "a"), "a",\n new Pair<>("e", "b"), "b",\n new Pair<>("a", "e"), "a",\n new Pair<>("a", "a"), "e",\n new Pair<>("a", "b"), "b",\n new Pair<>("b", "e"), "b",\n new Pair<>("b", "a"), "b",\n new Pair<>("b", "b"), "e"\n);\n\nProbableGroup group = new ProbableGroup(elements, opTable);\nSystem.out.println("Is group: " + group.isGroup()); \/\/ Expected: true<\/code><\/pre>\n
\nCode<\/h2>\n
package groups;\n\nimport java.util.Map;\nimport java.util.Set;\n\npublic final class ProbableGroup {\n\n private final Set<String> elements;\n private final Map<Pair<String>, String> opTable;\n\n public ProbableGroup(Set<String> elements, Map<Pair<String>, String> opTable) {\n this.elements = java.util.Collections.unmodifiableSet(\n new java.util.LinkedHashSet<>(elements)\n );\n java.util.Map<Pair<String>, String> copy = new java.util.LinkedHashMap<>();\n for (Map.Entry<Pair<String>, String> e : opTable.entrySet()) {\n Pair<String> k = e.getKey();\n copy.put(new Pair<>(k.getElem1(), k.getElem2()), e.getValue());\n }\n this.opTable = java.util.Collections.unmodifiableMap(copy);\n }\n\n private static Pair<String> key(String a, String b) { return new Pair<>(a, b); }\n\n public Set<String> getElements() {\n return new java.util.LinkedHashSet<>(elements);\n }\n\n public Map<Pair<String>, String> getOpTable() {\n java.util.Map<Pair<String>, String> copy = new java.util.LinkedHashMap<>();\n for (Map.Entry<Pair<String>, String> e : opTable.entrySet()) {\n Pair<String> k = e.getKey();\n copy.put(new Pair<>(k.getElem1(), k.getElem2()), e.getValue());\n }\n return copy;\n }\n\n public String product(String a, String b) {\n String r = opTable.get(key(a, b));\n if (r == null) throw new IllegalArgumentException("No entry for (" + a + "," + b + ")");\n return r;\n }\n\n public String getIdentity() {\n for (String cand : elements) {\n boolean left = true;\n for (String x : elements) {\n if (!product(cand, x).equals(x)) { left = false; break; }\n }\n if (!left) continue;\n boolean right = true;\n for (String x : elements) {\n if (!product(x, cand).equals(x)) { right = false; break; }\n }\n if (right) return cand;\n }\n return "";\n }\n\n public String getInverse(String a) {\n String e = getIdentity();\n if (e.isEmpty()) return "";\n for (String b : elements) {\n if (product(a, b).equals(e) && product(b, a).equals(e)) return b;\n }\n return "";\n }\n\n public String power(String a, int n) {\n if (n <= 0) throw new IllegalArgumentException("power requires n > 0, got " + n);\n String acc = a;\n for (int i = 1; i < n; i++) acc = product(acc, a);\n return acc;\n }\n\n public int order(String a) {\n String e = getIdentity();\n if (e.isEmpty()) return Integer.MAX_VALUE;\n String acc = a;\n int limit = Math.max(1, elements.size());\n for (int m = 1; m <= limit; m++) {\n if (acc.equals(e)) return m;\n acc = product(acc, a);\n }\n for (int m = limit + 1; m <= limit + elements.size(); m++) {\n if (acc.equals(e)) return m;\n acc = product(acc, a);\n }\n return Integer.MAX_VALUE;\n }\n\n public boolean isGroup() {\n for (String a : elements) for (String b : elements) {\n String r = opTable.get(key(a, b));\n if (r == null || !elements.contains(r)) return false;\n }\n for (String a : elements)\n for (String b : elements)\n for (String c : elements) {\n String left = product(product(a, b), c);\n String right = product(a, product(b, c));\n if (!left.equals(right)) return false;\n }\n String e = getIdentity();\n if (e.isEmpty()) return false;\n for (String a : elements) if (getInverse(a).isEmpty()) return false;\n return true;\n }\n\n public boolean isCommutative() {\n for (String a : elements) for (String b : elements) {\n if (!product(a, b).equals(product(b, a))) return false;\n }\n return true;\n }\n\n public boolean isSubgroup(Set<String> h) {\n if (h == null || h.isEmpty()) return false;\n if (!isGroup()) return false;\n if (!elements.containsAll(h)) return false;\n\n String e = getIdentity();\n if (e.isEmpty() || !h.contains(e)) return false;\n\n for (String s : h) for (String t : h) {\n if (!h.contains(product(s, t))) return false;\n }\n for (String s : h) {\n String inv = getInverse(s);\n if (inv.isEmpty() || !h.contains(inv)) return false;\n }\n return true;\n }\n}<\/code><\/pre>\n
\nCopilot \u8aaa<\/h2>\n
ProbableGroup<\/code>\uff0c\u7528\u4f86\u8868\u793a\u4e00\u500b\u53ef\u80fd\u662f\u7fa4\uff08Group\uff09\u7684\u4ee3\u6578\u7d50\u69cb\u3002\u5b83\u4f7f\u7528\u96c6\u5408\u548c\u6620\u5c04\u4f86\u63cf\u8ff0\u5143\u7d20\u8207\u5176\u4e8c\u5143\u904b\u7b97\u8868\uff0c\u4e26\u63d0\u4f9b\u4e00\u7cfb\u5217\u65b9\u6cd5\u4f86\u6aa2\u67e5\u9019\u500b\u7d50\u69cb\u662f\u5426\u6eff\u8db3\u7fa4\u7684\u6027\u8cea\u3002\u4ee5\u4e0b\u662f\u8a73\u7d30\u89e3\u91cb\uff1a<\/p>\n\ud83e\udde9 \u985e\u5225\u7d50\u69cb\u8207\u6210\u54e1<\/h3>\n
private final Set<String> elements;\nprivate final Map<Pair<String>, String> opTable;<\/code><\/pre>\n\n
elements<\/code>\uff1a\u7fa4\u7684\u5143\u7d20\u96c6\u5408\u3002<\/li>\nopTable<\/code>\uff1a\u4e8c\u5143\u904b\u7b97\u8868\uff0c\u5b9a\u7fa9\u4e86\u6bcf\u5c0d\u5143\u7d20\u7684\u4e58\u7a4d\u7d50\u679c\u3002<\/li>\n<\/ul>\n\ud83c\udfd7\ufe0f \u5efa\u69cb\u5b50<\/h3>\n
public ProbableGroup(Set<String> elements, Map<Pair<String>, String> opTable)<\/code><\/pre>\n\n
unmodifiableSet<\/code> \u548c unmodifiableMap<\/code>\uff09\uff0c\u78ba\u4fdd\u5916\u90e8\u7121\u6cd5\u4fee\u6539\u3002<\/li>\n<\/ul>\n\ud83d\udd27 \u6838\u5fc3\u65b9\u6cd5\u89e3\u6790<\/h3>\n
product(String a, String b)<\/code><\/h4>\n\n
a<\/code> \u548c b<\/code> \u7684\u4e58\u7a4d\u3002<\/li>\ngetIdentity()<\/code><\/h4>\n\n
\n
x<\/code>\uff0c\u82e5 e * x = x<\/code> \u4e14 x * e = x<\/code>\uff0c\u5247 e<\/code> \u70ba\u55ae\u4f4d\u5143\u3002<\/li>\ngetInverse(String a)<\/code><\/h4>\n\n
a<\/code> \u7684\u53cd\u5143\u7d20 b<\/code>\uff0c\u4f7f\u5f97 a * b = e<\/code> \u4e14 b * a = e<\/code>\u3002<\/li>\npower(String a, int n)<\/code><\/h4>\n\n
a^n<\/code>\uff0c\u5373 a<\/code> \u81ea\u4e58 n<\/code> \u6b21\u3002<\/li>\nn<\/code> \u5fc5\u9808\u5927\u65bc 0\u3002<\/li>\n<\/ul>\norder(String a)<\/code><\/h4>\n\n
a<\/code> \u7684\u968e\uff08order\uff09\uff0c\u5373\u6700\u5c0f\u7684\u6b63\u6574\u6578 m<\/code> \u4f7f\u5f97 a^m = e<\/code>\u3002<\/li>\n