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Group Action

In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

group action

A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K.

The symmetric group Sn acts on any set with n elements by permuting the elements of the set. Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality.

The action of G \displaystyle G on X \displaystyle X is called primitive if there is no partition of X \displaystyle X preserved by all elements of G \displaystyle G apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).

If g \displaystyle g acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero g \displaystyle g -invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.

The set of all orbits of X under the action of G is written as X/G (or, less frequently: G\X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written X G , \displaystyle X_G, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

Every orbit is an invariant subset of X on which G acts transitively. Conversely, any invariant subset of X is a union of orbits. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of G (that is, the set of all conjugates of the subgroup). Let ( H ) \displaystyle (H) denote the conjugacy class of H. Then the orbit O has type ( H ) \displaystyle (H) if the stabilizer G x \displaystyle G_x of some/any x in O belongs to ( H ) \displaystyle (H) . A maximal orbit type is often called a principal orbit type.

where Xg is the set of points fixed by g. This result is mainly of use when G and X are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

Fixing a group G, the set of formal differences of finite G-sets forms a ring called the Burnside ring of G, where addition corresponds to disjoint union, and multiplication to Cartesian product.

Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.

We can view a group G as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.

In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.

When Azure Monitor data indicates that there might be a problem with your infrastructure or application, an alert is triggered. Azure Monitor, Azure Service Health, and Azure Advisor then use action groups to notify users about the alert and take an action. An action group is a collection of notification preferences that are defined by the owner of an Azure subscription.

This article shows you how to create and manage action groups in the Azure portal. Depending on your requirements, you can configure various alerts to use the same action group or different action groups.

An action group is a global service, so there's no dependency on a specific Azure region. Requests from clients can be processed by action group services in any region. For instance, if one region of the action group service is down, the traffic is automatically routed and processed by other regions. As a global service, an action group helps provide a disaster recovery solution.

Under Instance details, enter values for Action group name and Display name. The display name is used in place of a full action group name when the group is used to send notifications.

Details: Enter appropriate information for your selected action type. For instance, you might enter a webhook URI, the name of an Azure app, an ITSM connection, or an Automation runbook. For an ITSM action, also enter values for Work item and other fields that your ITSM tool requires.

To assign a key-value pair to the action group, select Next: Tags. Alternately, at the top of the page, select the Tags tab. Otherwise, skip this step. By using tags, you can categorize your Azure resources. Tags are available for all Azure resources, resource groups, and subscriptions.

To review your settings, select Review + create. This step quickly checks your inputs to make sure you've entered all required information. If there are issues, they're reported here. After you've reviewed the settings, select Create to create the action group.

When you run a test and select a notification type, you get a message with "Test" in the subject. The tests provide a way to check that your action group works as expected before you enable it in a production environment. All the details and links in test email notifications are from a sample reference set.

You can use an Azure Resource Manager template to configure action groups. Using templates, you can automatically set up action groups that can be reused in certain types of alerts. These action groups ensure that all the correct parties are notified when an alert is triggered.

To create an action group by using a Resource Manager template, you create a resource of the type Microsoft.Insights/actionGroups. Then you fill in all related properties. Here are two sample templates that create an action group.

The first template describes how to create a Resource Manager template for an action group where the action definitions are hard-coded in the template. The second template describes how to create a template that takes the webhook configuration information as input parameters when the template is deployed.

You might have a limited number of email actions per action group. For information about rate limits, see Rate limiting for voice, SMS, emails, Azure App Service push notifications, and webhook posts.

When you use this type of notification, you can send email to the members of a subscription's role. Email is only sent to Azure Active Directory (Azure AD) user members of the role. Email isn't sent to Azure AD groups or service principals.

You might have a limited number of email actions per action group. To check which limits apply to your situation, see Rate limiting for voice, SMS, emails, Azure App Service push notifications, and webhook posts.

An action that uses Functions calls an existing HTTP trigger endpoint in Functions. For more information about Functions, see Azure Functions. To handle a request, your endpoint must handle the HTTP POST verb.

When you define the function action, the function's HTTP trigger endpoint and access key are saved in the action definition, for example, =. If you change the access key for the function, you must remove and re-create the function action in the action group.

The secure webhook action authenticates to the protected API by using a Service Principal instance in the Azure AD tenant of the "AZNS AAD Webhook" Azure AD application. To make the action group work, this Azure AD Webhook Service Principal must be added as a member of a role on the target Azure AD application that grants access to the target endpoint.

If you use the webhook action, your target webhook endpoint must be able to process the various JSON payloads that different alert sources emit. If the webhook endpoint expects a specific schema, for example, the Microsoft Teams schema, use the Logic Apps action to transform the alert schema to meet the target webhook's expectations. 041b061a72

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